Pascal's triangle - Knowledge

8916:. This process of summing the number of elements of a given dimension to those of one fewer dimension to arrive at the number of the former found in the next higher simplex is equivalent to the process of summing two adjacent numbers in a row of Pascal's triangle to yield the number below. Thus, the meaning of the final number (1) in a row of Pascal's triangle becomes understood as representing the new vertex that is to be added to the simplex represented by that row to yield the next higher simplex represented by the next row. This new vertex is joined to every element in the original simplex to yield a new element of one higher dimension in the new simplex, and this is the origin of the pattern found to be identical to that seen in Pascal's triangle. 9318: 9078: 2612: 9313:{\displaystyle {\begin{matrix}{\text{ 1}}\\{\text{ 1}}\quad {\text{ 2}}\\{\text{ 1}}\quad {\text{ 4}}\quad {\text{ 4}}\\{\text{ 1}}\quad {\text{ 6}}\quad {\text{ 12}}\quad {\text{ 8}}\\{\text{ 1}}\quad {\text{ 8}}\quad {\text{ 24}}\quad {\text{ 32}}\quad {\text{ 16}}\\{\text{ 1}}\quad {\text{ 10}}\quad {\text{ 40}}\quad {\text{ 80}}\quad {\text{ 80}}\quad {\text{ 32}}\\{\text{ 1}}\quad {\text{ 12}}\quad {\text{ 60}}\quad 160\quad 240\quad 192\quad {\text{ 64}}\\{\text{ 1}}\quad {\text{ 14}}\quad {\text{ 84}}\quad 280\quad 560\quad 672\quad 448\quad 128\end{matrix}}} 8638: 2053: 6036: 810: 4564: 4572: 3263: 8125: 9336:) can be read from the table in a way analogous to Pascal's triangle. For example, the number of 2-dimensional elements in a 2-dimensional cube (a square) is one, the number of 1-dimensional elements (sides, or lines) is 4, and the number of 0-dimensional elements (points, or vertices) is 4. This matches the 2nd row of the table (1, 4, 4). A cube has 1 cube, 6 faces, 12 edges, and 8 vertices, which corresponds to the next line of the analog triangle (1, 6, 12, 8). This pattern continues indefinitely. 7951: 8285: 2607:{\displaystyle {\begin{array}{c}{\dbinom {0}{0}}\\{\dbinom {1}{0}}\quad {\dbinom {1}{1}}\\{\dbinom {2}{0}}\quad {\dbinom {2}{1}}\quad {\dbinom {2}{2}}\\{\dbinom {3}{0}}\quad {\dbinom {3}{1}}\quad {\dbinom {3}{2}}\quad {\dbinom {3}{3}}\\{\dbinom {4}{0}}\quad {\dbinom {4}{1}}\quad {\dbinom {4}{2}}\quad {\dbinom {4}{3}}\quad {\dbinom {4}{4}}\\{\dbinom {5}{0}}\quad {\dbinom {5}{1}}\quad {\dbinom {5}{2}}\quad {\dbinom {5}{3}}\quad {\dbinom {5}{4}}\quad {\dbinom {5}{5}}\end{array}}} 13306: 7935: 2656: 8730: 632: 7995: 8633:{\displaystyle {\begin{aligned}\exp {\begin{pmatrix}.&.&.&.&.\\1&.&.&.&.\\.&2&.&.&.\\.&.&3&.&.\\.&.&.&4&.\end{pmatrix}}&={\begin{pmatrix}1&.&.&.&.\\1&1&.&.&.\\1&2&1&.&.\\1&3&3&1&.\\1&4&6&4&1\end{pmatrix}}\\e^{\text{counting}}&={\text{binomial}}\end{aligned}}} 655: 8062: 8039: 8090: 8046: 8083: 8069: 8055: 8032: 8025: 8016: 8009: 8002: 207: 13316: 3258:{\displaystyle {\begin{aligned}\sum _{i=0}^{n}a_{i}x^{i+1}+\sum _{k=0}^{n}a_{k}x^{k}&=\sum _{k=1}^{n+1}a_{k-1}x^{k}+\sum _{k=0}^{n}a_{k}x^{k}\\&=\sum _{k=1}^{n}a_{k-1}x^{k}+a_{n}x^{n+1}+a_{0}x^{0}+\sum _{k=1}^{n}a_{k}x^{k}\\&=a_{0}x^{0}+\sum _{k=1}^{n}(a_{k-1}+a_{k})x^{k}+a_{n}x^{n+1}\\&=x^{0}+\sum _{k=1}^{n}(a_{k-1}+a_{k})x^{k}+x^{n+1}.\end{aligned}}} 8835:-simplex consists of simply adding a new vertex to the latter, positioned such that this new vertex lies outside of the space of the original simplex, and connecting it to all original vertices. As an example, consider the case of building a tetrahedron from a triangle, the latter of whose elements are enumerated by row 3 of Pascal's triangle: 23: 5089: 309: 6354: 6026:: When the elements of a row of Pascal's triangle are alternately added and subtracted together, the result is 0. For example, row 6 is 1, 6, 15, 20, 15, 6, 1, so the formula is 1 − 6 + 15 − 20 + 15 − 6 + 1 = 0. 11287:

from its leftmost digit up to, but excluding, its rightmost digit, and use radix-twelve arithmetic to sum the removed prefix with the entry on its immediate left, then repeat this process, proceeding leftward, until the leftmost entry is reached. In this particular example, the normalized string ends

299:

and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number

4597:

The sum of the elements of a single row is twice the sum of the row preceding it. For example, row 0 (the topmost row) has a value of 1, row 1 has a value of 2, row 2 has a value of 4, and so forth. This is because every item in a row produces two items in the next row: one left and

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The other way of producing this triangle is to start with Pascal's triangle and multiply each entry by 2, where k is the position in the row of the given number. For example, the 2nd value in row 4 of Pascal's triangle is 6 (the slope of 1s corresponds to the zeroth entry in each row). To get the

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Pascal's triangle. Rather than performing the multiplicative calculation, one can simply look up the appropriate entry in the triangle (constructed by additions). For example, suppose 3 workers need to be hired from among 7 candidates; then the number of possible hiring choices is 7 choose 3, the

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has one 3-dimensional element (itself), four 2-dimensional elements (faces), six 1-dimensional elements (edges), and four 0-dimensional elements (vertices). Adding the final 1 again, these values correspond to the 4th row of the triangle (1, 4, 6, 4, 1). Line 1 corresponds to a point, and Line 2

9946:

once observed that the first five rows of Pascal's triangle, when read as the digits of an integer, are the corresponding powers of eleven. He claimed without proof that subsequent rows also generate powers of eleven. In 1964, Robert L. Morton presented the more generalized argument that each

4855: 1444: 11734: 202:{\displaystyle {\begin{array}{c}1\\1\quad 1\\1\quad 2\quad 1\\1\quad 3\quad 3\quad 1\\1\quad 4\quad 6\quad 4\quad 1\\1\quad 5\quad 10\quad 10\quad 5\quad 1\\1\quad 6\quad 15\quad 20\quad 15\quad 6\quad 1\\1\quad 7\quad 21\quad 35\quad 35\quad 21\quad 7\quad 1\end{array}}} 3867: 7834: 2045: 4547:

independent copies of that variable; this is exactly the situation to which the central limit theorem applies, and hence results in the normal distribution in the limit. (The operation of repeatedly taking a convolution of something with itself is called the

5512: 983: 6542: 4218: 5213: 8693:

can be used to prove the geometric relationship provided by Pascal's triangle. This same proof could be applied to simplices except that the first column of all 1's must be ignored whereas in the algebra these correspond to the real numbers,

11173: 9364:-cube, one must double the first of a pair of numbers in a row of this analog of Pascal's triangle before summing to yield the number below. The initial doubling thus yields the number of "original" elements to be found in the next higher 6089: 9368:-cube and, as before, new elements are built upon those of one fewer dimension (edges upon vertices, faces upon edges, etc.). Again, the last number of a row represents the number of new vertices to be added to generate the next higher 10390: 4850: 8806:

Let's begin by considering the 3rd line of Pascal's triangle, with values 1, 3, 3, 1. A 2-dimensional triangle has one 2-dimensional element (itself), three 1-dimensional elements (lines, or edges), and three 0-dimensional elements

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to the space of the original figure, then connecting each vertex of the new figure to its corresponding vertex of the original. This initial duplication process is the reason why, to enumerate the dimensional elements of an

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above and to the right, treating blank entries as 0. For example, the initial number of row 1 (or any other row) is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in row 3 are added to produce the number 4 in row 4.

7570: 6796: 9067: 7679: 6948: 5306: 2661: 3657: 9698: 9607: 7929: 10980: 7971:

the Sierpinski triangle, assuming a fixed perimeter. More generally, numbers could be colored differently according to whether or not they are multiples of 3, 4, etc.; this results in other similar patterns.

5636: 565: 5776: 1815: 7279: 7211: 11622: 5924: 8116:

In a triangular portion of a grid (as in the images below), the number of shortest grid paths from a given node to the top node of the triangle is the corresponding entry in Pascal's triangle. On a

5339: 5084:{\displaystyle {\frac {s_{n+1}}{s_{n}}}={\frac {\displaystyle (n+1)!^{n+2}\prod _{k=0}^{n+1}{\frac {1}{k!^{2}}}}{\displaystyle n!^{n+1}\prod _{k=0}^{n}{\frac {1}{k!^{2}}}}}={\frac {(n+1)^{n}}{n!}}} 3617: 824: 10719: 1810: 8290: 6094: 7684: 8850:

The number of a given dimensional element in the tetrahedron is now the sum of two numbers: first the number of that element found in the original triangle, plus the number of new elements,

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Each frame represents a row in Pascal's triangle. Each column of pixels is a number in binary with the least significant bit at the bottom. Light pixels represent 1 and dark pixels 0.

11839: 9781: 10579: 10119: 1546: 4310: 10876: 7449: 7143: 6843: 1124: 1077: 1030: 450: 11591: 8847:

vertices. To build a tetrahedron from a triangle, position a new vertex above the plane of the triangle and connect this vertex to all three vertices of the original triangle.

4404: 10831: 7614: 395: 11285: 9868: 7096: 7067: 7038: 7009: 6980: 7954:

A level-4 approximation to a Sierpinski triangle obtained by shading the first 32 rows of a Pascal triangle white if the binomial coefficient is even and black if it is odd.

4076: 3421: 3303: 617: 9925: 9890: 8714: 11539: 8944:. There are a couple ways to do this. The simpler is to begin with row 0 = 1 and row 1 = 1, 2. Proceed to construct the analog triangles according to the following rule: 5821: 3375: 1659: 11477: 11013: 10432: 10174: 6349:{\displaystyle {\begin{aligned}P_{0}(n)&=P_{d}(0)=1,\\P_{d}(n)&=P_{d}(n-1)+P_{d-1}(n)\\&=\sum _{i=0}^{n}P_{d-1}(i)=\sum _{i=0}^{d}P_{i}(n-1).\end{aligned}}} 3520: 1698: 9414: 5965: 1611: 11042: 4715: 3454: 12799: 12772: 12745: 12718: 12691: 11419: 11372: 10637: 11208: 10611: 9489: − 1) are the same, except that the sign alternates from +1 to −1 and back again. After suitable normalization, the same pattern of numbers occurs in the 4689: 4643: 4345: 3956: 3652: 3481: 3330: 2651: 1575: 1471: 11611: 5667: 5329: 12666:

But these in the alternate areas, which are given, I observed were the same with the figures of which the several ascending powers of the number 11 consist, viz.

11507: 11445: 11234: 10902: 10478: 10248: 4537: 1724: 1616:

To see how the binomial theorem relates to the simple construction of Pascal's triangle, consider the problem of calculating the coefficients of the expansion of

1161: 297: 271: 11306: 10121:. More rigorous proofs have since been developed. To better understand the principle behind this interpretation, here are some things to recall about binomials: 11559: 11392: 11346: 11326: 11034: 10786: 10766: 10739: 10522: 10498: 10452: 10273: 10268: 10218: 10194: 10143: 10062: 9988: 9968: 6695: 5234: 4720: 4616: 4432: 4365: 4262: 4242: 4068: 4048: 4028: 4008: 3976: 3929: 3909: 3889: 1491: 1181: 585: 354: 334: 7967:. This resemblance becomes increasingly accurate as more rows are considered; in the limit, as the number of rows approaches infinity, the resulting pattern 5239: 777:) was published posthumously in 1665. In this, Pascal collected several results then known about the triangle, and employed them to solve problems in 8109:

Pascal's triangle overlaid on a grid gives the number of distinct paths to each square, assuming only rightward and downward movements are considered.

6845:. For each subsequent element, the value is determined by multiplying the previous value by a fraction with slowly changing numerator and denominator: 6626:) then equals the total number of dots in the shape. A 0-dimensional triangle is a point and a 1-dimensional triangle is simply a line, and therefore 5541: 11797: 8657:

can be given: Pascal's triangle is the exponential of the matrix which has the sequence 1, 2, 3, 4, ... on its subdiagonal and zero everywhere else.

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rather than matrices. Recognising the geometric operations, such as rotations, allows the algebra operations to be discovered. Just as each row,

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From Alexandria, Through Baghdad: Surveys and Studies in the Ancient Greek and Medieval Islamic Mathematical Sciences in Honor of J.L. Berggren

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Pascal's triangle may be extended upwards, above the 1 at the apex, preserving the additive property, but there is more than one way to do so.

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That is, choose a pair of numbers according to the rules of Pascal's triangle, but double the one on the left before adding. This results in:

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to 2 or to 3 mod 4, then the signs start with −1. In fact, the sequence of the (normalized) first terms corresponds to the powers of

1729: 8928:, as opposed to triangles. To find the pattern, one must construct an analog to Pascal's triangle, whose entries are the coefficients of 1439:{\displaystyle (x+y)^{n}=\sum _{k=0}^{n}a_{k}x^{n-k}y^{k}=a_{0}x^{n}+a_{1}x^{n-1}y+a_{2}x^{n-2}y^{2}+\ldots +a_{n-1}xy^{n-1}+a_{n}y^{n},} 9622: 9531: 7839: 10907: 6062:

in order. The 1-dimensional simplex numbers increment by 1 as the line segments extend to the next whole number along the number line.

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in two ways. First, polynomial multiplication corresponds exactly to discrete convolution, so that repeatedly convolving the sequence

9328:. Now that the analog triangle has been constructed, the number of elements of any dimension that compose an arbitrarily dimensioned 4655: 9612:

compose the 4th row of the triangle, with alternating signs. This is a generalization of the following basic result (often used in

5676: 12010: 11729:{\displaystyle 1.1_{1234}^{1234}=2.885:2:35:977:696:\overbrace {\ldots } ^{\text{1227 digits}}:0:1_{1234}=2.717181235\ldots _{10}} 13165: 7216: 8811:, or corners). The meaning of the final number (1) is more difficult to explain (but see below). Continuing with our example, a 7148: 13340: 3862:{\displaystyle \sum _{k=0}^{n}{n \choose k}={n \choose 0}+{n \choose 1}+\cdots +{n \choose n-1}+{n \choose n}=(1+1)^{n}=2^{n}.} 12057: 5845: 12114: 11935: 8751: 6674:

There are simple algorithms to compute all the elements in a row or diagonal without computing other elements or factorials.

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published a portion of the triangle (from the second to the middle column in each row) in 1544, describing it as a table of

13131: 11561:(the number of places to the left the decimal has moved) is one less than the row length. Below is the normalized value of 10646: 8816:

corresponds to a line segment (dyad). This pattern continues to arbitrarily high-dimensioned hyper-tetrahedrons (known as

1496: 7829:{\displaystyle {\tbinom {5}{0}}=1,{\tbinom {6}{1}}=1\times {\tfrac {6}{1}}=6,{\tbinom {7}{2}}=6\times {\tfrac {7}{2}}=21} 9428:-dimensional cube. For example, in three dimensions, the third row (1 3 3 1) corresponds to the usual three-dimensional 12801:, etc. that is, first 1; the second 1, 1; the third 1, 2, 1; the fourth 1, 3, 3, 1; the fifth 1, 4, 6, 4, 1, and so on 2040:{\displaystyle (x+1)^{n+1}=(x+1)(x+1)^{n}=x(x+1)^{n}+(x+1)^{n}=\sum _{i=0}^{n}a_{i}x^{i+1}+\sum _{k=0}^{n}a_{k}x^{k}.} 12967: 12639: 12601: 12564: 12230: 12194: 12067: 12028: 11868: 8777: 4448: 9994: 8759: 665:

The pattern of numbers that forms Pascal's triangle was known well before Pascal's time. The Persian mathematician

643: 4539:, and hence to generating the rows of the triangle. Second, repeatedly convolving the distribution function for a 8120:

game board shaped like a triangle, this distribution should give the probabilities of winning the various prizes.

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Pascal's triangle was known in China during the early 11th century through the work of the Chinese mathematician

11953: 5507:{\displaystyle \pi =3+\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {2n+1 \choose 1}{{2n+1 \choose 2}{2n+2 \choose 2}}}} 13319: 9525:

th row of the triangle with alternating signs. For example, the values of the step function that results from:

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Due to its simple construction by factorials, a very basic representation of Pascal's triangle in terms of the

978:{\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}=\mathbf {1} x^{2}y^{0}+\mathbf {2} x^{1}y^{1}+\mathbf {1} x^{0}y^{2},} 13350: 13097: 12398: 11782: 12225: 11240:

the numeral, simply carry the first compound entry's prefix, that is, remove the prefix of the coefficient

6602:(1) = 1. Place these dots in a manner analogous to the placement of numbers in Pascal's triangle. To find P 12626:. In International Series in Modern Applied Mathematics and Computer Science. Pergamon. pp. 100–102. 10527: 10067: 764:

also published the triangle as well as the additive and multiplicative rules for constructing it in 1570.

13158: 13092: 12947: 12619: 12588:. In International Series in Modern Applied Mathematics and Computer Science. Pergamon. pp. 89–102. 12581: 11613: 4576: 4268: 12954:. In International Series in Modern Applied Mathematics and Computer Science. Pergamon. pp. 89–91. 10836: 8132:

If the rows of Pascal's triangle are left-justified, the diagonal bands (colour-coded below) sum to the

7407: 7101: 6801: 1082: 1035: 988: 408: 13127:(from the Ssu Yuan Yü Chien of Chu Shi-Chieh, 1303, depicting the first nine rows of Pascal's triangle) 11564: 9351:-cube is done by simply duplicating the original figure and displacing it some distance (for a regular 6537:{\displaystyle P_{d}(n)={\frac {1}{d!}}\prod _{k=0}^{d-1}(n+k)={n^{(d)} \over d!}={\binom {n+d-1}{d}},} 4374: 4213:{\displaystyle \mathbf {C} (n,k)=\mathbf {C} _{k}^{n}={_{n}C_{k}}={n \choose k}={\frac {n!}{k!(n-k)!}}} 782: 10791: 9424:

Each row of Pascal's triangle gives the number of vertices at each distance from a fixed vertex in an

7578: 359: 13087: 11792: 11243: 9991: 9829: 7072: 7043: 7014: 6985: 6956: 5216: 5208:{\displaystyle {\frac {s_{n+1}\cdot s_{n-1}}{s_{n}^{2}}}=\left({\frac {n+1}{n}}\right)^{n},~n\geq 1.} 4660: 12333: 8740: 3380: 3270: 590: 12167:

Edwards, A. W. F. (2013), "The arithmetical triangle", in Wilson, Robin; Watkins, John J. (eds.),

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To understand why this pattern exists, one must first understand that the process of building an

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The pattern obtained by coloring only the odd numbers in Pascal's triangle closely resembles the

5781: 3335: 1619: 12266: 11450: 11168:{\displaystyle 11_{12}^{12}=1:10:56:164:353:560:650:560:353:164:56:10:1_{12}=27433a9699701_{12}} 10986: 10741: 13345: 13309: 13197: 13151: 10398: 10152: 9613: 3530: 3486: 1664: 13123: 12554: 12431:. See in particular Theorem 2, which gives a generalization of this fact for all prime moduli. 10064:, when interpreted directly as a place-value numeral, correspond to the binomial expansion of 9386: 5929: 1580: 12913: 12184: 12043: 11925: 11824: 11787: 11236:

are compound because these row entries compute to values greater than or equal to twelve. To

10983: 9718:, which cycle around the intersection of the axes with the unit circle in the complex plane: 9707:. The corresponding row of the triangle is row 0, which consists of just the number 1. 8808: 5998: 4694: 4435: 4407: 4368: 3426: 1577:

in these binomial expansions, while the next left diagonal corresponds to the coefficient of

1134: 12777: 12750: 12723: 12696: 12669: 11397: 11351: 10616: 6652:, which is the sequence of natural numbers. The number of dots in each layer corresponds to 273:

at the top (the 0th row). The entries in each row are numbered from the left beginning with

232:

which play a crucial role in probability theory, combinatorics, and algebra. In much of the

13234: 12851:"Newton's Unfinished Business: Uncovering the Hidden Powers of Eleven in Pascal's Triangle" 12527: 12518:

Hore, P. J. (1983), "Solvent suppression in Fourier transform nuclear magnetic resonance",

12470: 12427: 12292: 12209: 12016: 11897: 11181: 10584: 9928: 4667: 4621: 4563: 4442: 4323: 3934: 3625: 3459: 3308: 2624: 1553: 1449: 670: 229: 13288: 11596: 10385:{\displaystyle 14641_{a}=1\cdot a^{4}+4\cdot a^{3}+6\cdot a^{2}+4\cdot a^{1}+1\cdot a^{0}} 5643: 5314: 4845:{\displaystyle s_{n}=\prod _{k=0}^{n}{n \choose k}=\prod _{k=0}^{n}{\frac {n!}{k!(n-k)!}}} 801:(Latin: Pascal's Arithmetic Triangle), which became the basis of the modern Western name. 8: 13207: 12370: 12210:

Traité du triangle arithmétique, avec quelques autres petits traitez sur la mesme matière

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Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures

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th power of 2. This is equivalent to the statement that the number of subsets of an

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An interesting consequence of the binomial theorem is obtained by setting both variables

3526: 1703: 1140: 276: 250: 12531: 12020: 11291: 8893: 13036: 13001: 12959: 12894: 12831: 12631: 12593: 12458: 12415: 12350: 12309: 12247: 12088: 11885: 11834: 11762: 11544: 11377: 11331: 11311: 11019: 10771: 10751: 10724: 10507: 10483: 10437: 10253: 10203: 10179: 10128: 10047: 10044:, of the triangle, and the rows are its partial products. He proved the entries of row 9973: 9953: 9324:

value that resides in the corresponding position in the analog triangle, multiply 6 by

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be the number of 1s in the binary representation. Then the number of odd terms will be

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remains in the numerator after integer division, making the entire entry a multiple of

5219: 4601: 4417: 4350: 4247: 4227: 4053: 4033: 4013: 3993: 3961: 3914: 3894: 3874: 1476: 1166: 818: 778: 729: 570: 339: 319: 13280: 8852:

each of which is built upon elements of one fewer dimension from the original triangle

13272: 13217: 13109: 13106: 13068: 12963: 12869: 12635: 12597: 12560: 12539: 12354: 12313: 12190: 12145: 12128: 12110: 12063: 12024: 11931: 11767: 11237: 9490: 8925: 8686: 8670: 6066: 4549: 792: 741: 677:(1048–1131), another Persian mathematician; thus the triangle is also referred to as 13264: 9462:). The second row corresponds to a square, while larger-numbered rows correspond to 7994: 5778:. For example, in row 4, which is 1, 4, 6, 4, 1, we get the 3rd Catalan number 5215:

The right-hand side of the above equation takes the form of the limit definition of

13063: 13028: 12993: 12955: 12928: 12823: 12627: 12589: 12535: 12450: 12407: 12383: 12379: 12342: 12301: 12239: 12140: 11877: 10041: 9339:

To understand why this pattern exists, first recognize that the construction of an

8690: 8666: 8133: 7980:, a corollary is that the proportion of odd binomial coefficients tends to zero as 6549: 6077: 4648:

Taking the product of the elements in each row, the sequence of products (sequence

1130: 761: 757: 688: 620: 452:. With this notation, the construction of the previous paragraph may be written as 225: 12475: 6035: 312:

In Pascal's triangle, each number is the sum of the two numbers directly above it.

13202: 12466: 12423: 11893: 11863: 9704: 9329: 8681:-simplex, as described below, it also defines the number of named basis forms in 4540: 809: 749: 12044:

The Development of Arabic Mathematics Between Arithmetic and Algebra - R. Rashed

3522:. This is indeed the downward-addition rule for constructing Pascal's triangle. 1550:

The entire left diagonal of Pascal's triangle corresponds to the coefficient of

674: 11772: 9715: 9514: 8061: 8038: 7934: 7397:{\displaystyle {\tbinom {n}{0}},{\tbinom {n+1}{1}},{\tbinom {n+2}{2}},\ldots ,} 6059: 5670: 4571: 745: 737: 12346: 8124: 8089: 8045: 4543:

with itself corresponds to calculating the distribution function for a sum of

687:) in Iran. Several theorems related to the triangle were known, including the 669:(953–1029) wrote a now-lost book which contained the first formulation of the 247:

The rows of Pascal's triangle are conventionally enumerated starting with row

13334: 13174: 12305: 11803: 11752: 10640: 9932: 9518: 8648: 8082: 8068: 8054: 8031: 8024: 8015: 8008: 8001: 7950: 4560:

Pascal's triangle has many properties and contains many patterns of numbers.

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based on the binomial expansion, and therefore on the binomial coefficients.

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studied it centuries before him in Persia, India, China, Germany, and Italy.

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Thus the extreme left and right coefficients remain as 1, and for any given

11746: 10221: 9943: 8864: 8791: 7565:{\displaystyle {n+k \choose k}={n+k-1 \choose k-1}\times {\frac {n+k}{k}}.} 5835: 12984:

Mueller, Francis J. (1965), "More on Pascal's Triangle and powers of 11",

10197: 9485: + 1) are the nth row of the triangle. Now the coefficients of ( 6791:{\displaystyle {\tbinom {n}{0}},{\tbinom {n}{1}},\ldots ,{\tbinom {n}{n}}} 3986:

A second useful application of Pascal's triangle is in the calculation of

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during the early 14th century, using the multiplicative formula for them.

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Finding any row of Pascal's triangle extending the concept of power of 11

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with compound digits (delimited by ":") in radix twelve. The digits from

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the coefficients are the entries in the second row of Pascal's triangle:

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and the first description of Pascal's triangle. It was later repeated by

217: 13040: 13005: 12898: 12835: 9062:{\displaystyle {n \choose k}=2\times {n-1 \choose k-1}+{n-1 \choose k}.} 3978:

elements may be independently included or excluded from a given subset.

12462: 12419: 12251: 11889: 11818: 10504:. Thus, when the entries of the row are concatenated and read in radix 9356: 8665:

Labelling the elements of each n-simplex matches the basis elements of

7674:{\displaystyle {\tfrac {6}{1}},{\tfrac {7}{2}},{\tfrac {8}{3}},\ldots } 7451:

and obtain subsequent elements by multiplication by certain fractions:

6943:{\displaystyle {n \choose k}={n \choose k-1}\times {\frac {n+1-k}{k}}.} 5986: 5301:{\displaystyle e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}} 3619:, the coefficients are identical in the expansion of the general case. 733: 647: 13244: 13019:

Low, Leone (1966), "Even more on Pascal's Triangle and Powers of 11",

12948:"Extending the binomial coefficients to preserve symmetry and pattern" 12620:"Extending the binomial coefficients to preserve symmetry and pattern" 12582:"Extending the binomial coefficients to preserve symmetry and pattern" 7281:, etc. (The remaining elements are most easily obtained by symmetry.) 13114: 12932: 12657: 12502: 12441:

Hinz, Andreas M. (1992), "Pascal's triangle and the Tower of Hanoi",

11480: 9799: 9506: 9463: 9333: 4315: 666: 12914:"A generalization of Pascal's triangle using powers of base numbers" 12454: 12411: 12243: 11881: 8729: 6593:) by placing additional dots below an initial dot, corresponding to 6055:

The diagonals going along the left and right edges contain only 1's.

12885:

Winteridge, David J. (1984), "Pascal's Triangle and Powers of 11",

12491:

Ian Stewart, "How to Cut a Cake", Oxford University Press, page 180

12368:

Foster, T. (2014), "Nilakantha's Footprints in Pascal's Triangle",

8795: 8660: 6586: 706: 702: 692: 635: 631: 9693:{\displaystyle {\mathfrak {Re}}\left({\text{Fourier}}\left\right)} 9602:{\displaystyle {\mathfrak {Re}}\left({\text{Fourier}}\left\right)} 9444:

itself), three vertices at distance 1, three vertices at distance

9379:

is equal to 3. Again, to use the elements of row 4 as an example:

7924:{\displaystyle {\tbinom {5}{5}},{\tbinom {6}{5}},{\tbinom {7}{5}}} 654: 12213: 12087:. National Council of Teachers of Mathematics. pp. 140–142. 11840:

Polynomials calculating sums of powers of arithmetic progressions

10975:{\displaystyle n^{n}\left(1+{\frac {1}{n}}\right)^{n}=11_{n}^{n}} 8817: 7976:

As the proportion of black numbers tends to zero with increasing

7960: 6040: 5516:

Some of the numbers in Pascal's triangle correlate to numbers in

13143: 12814:

Morton, Robert L. (1964), "Pascal's Triangle and powers of 11",

8794:

for the number of elements (such as edges and corners) within a

8642:

Binomial matrix as matrix exponential. All the dots represent 0.

5669:, the middle term minus the term two spots to the left equals a 5631:{\displaystyle \sum _{k=0}^{n}{n \choose k}^{2}={2n \choose n}.} 724:

In Europe, Pascal's triangle appeared for the first time in the

8117: 560:{\displaystyle {n \choose k}={n-1 \choose k-1}+{n-1 \choose k}} 13104: 11593:. Compound digits remain in the value because they are radix 5842:. This can be proven easily, from the multiplicative formula 5771:{\displaystyle C_{m-1}={\tbinom {2m}{m}}-{\tbinom {2m}{m-2}}} 817:

Pascal's triangle determines the coefficients which arise in

760:(1500–1577), who published six rows of the triangle in 1556. 732:(13th century). The binomial coefficients were calculated by 619:. This recurrence for the binomial coefficients is known as 12396:

Fine, N. J. (1947), "Binomial coefficients modulo a prime",

12327:

Brothers, H. J. (2012), "Pascal's triangle: The hidden stor-

8860: 4438:

to the factorials involved in the formula for combinations.

211:

A diagram showing the first eight rows of Pascal's triangle.

16:

Triangular array of the binomial coefficients in mathematics

9870:, Pascal's triangle can be extended beyond the integers to 9802:

generalizations. The three-dimensional version is known as

9429: 7274:{\displaystyle {\tbinom {5}{2}}=5\times {\tfrac {4}{2}}=10} 5838:, all the terms in that row except the 1s are divisible by 4650: 12658:"A Treatise of the Method of Fluxions and Infinite Series" 12290:

Brothers, H. J. (2012), "Finding e in Pascal's triangle",

10480:. The expansion now typifies the expanded form of a radix 9419: 8798:(such as a triangle, a tetrahedron, a square, or a cube). 7206:{\displaystyle {\tbinom {5}{1}}=1\times {\tfrac {5}{1}}=5} 6669: 6375:

An alternative formula that does not involve recursion is

6953:

For example, to calculate row 5, the fractions are

11447:. It follows that the length of the normalized value of 8677:, starting at 0, of Pascal's triangle corresponds to an 5919:{\displaystyle {\tbinom {p}{k}}={\tfrac {p!}{k!(p-k)!}}} 6065:

Moving inwards, the next pair of diagonals contain the

5311: 813:

Visualisation of binomial expansion up to the 4th power

12474:. Hinz attributes this observation to an 1891 book by 12015:. Springer Science & Business Media. p. 132. 11957: 9521:, whose values (suitably normalized) are given by the 9083: 8458: 8304: 7898: 7871: 7844: 7809: 7776: 7755: 7722: 7689: 7654: 7639: 7624: 7583: 7412: 7354: 7319: 7292: 7254: 7221: 7186: 7153: 7106: 7077: 7048: 7019: 6990: 6961: 6806: 6765: 6732: 6705: 5877: 5850: 5732: 5700: 4385: 4273: 3612:{\displaystyle (a+b)^{n}=b^{n}({\tfrac {a}{b}}+1)^{n}} 3582: 2058: 1087: 1040: 993: 413: 364: 28: 13124:

The Old Method Chart of the Seven Multiplying Squares

12780: 12753: 12726: 12699: 12672: 12228:(January 1996). "The Binomial Coefficient Function". 12062:. Springer Science & Business Media. p. 54. 11956: 11625: 11599: 11567: 11547: 11515: 11489: 11453: 11427: 11400: 11380: 11354: 11334: 11314: 11294: 11246: 11216: 11184: 11045: 11022: 10989: 10910: 10884: 10839: 10794: 10774: 10754: 10727: 10649: 10619: 10587: 10530: 10510: 10486: 10460: 10440: 10401: 10276: 10256: 10230: 10206: 10182: 10155: 10131: 10070: 10050: 9997: 9976: 9956: 9898: 9876: 9832: 9724: 9625: 9534: 9389: 9081: 8953: 8700: 8288: 8276: 7842: 7687: 7622: 7581: 7460: 7410: 7290: 7219: 7151: 7104: 7075: 7046: 7017: 6988: 6959: 6854: 6804: 6703: 6683: 6381: 6092: 6058:

The diagonals next to the edge diagonals contain the

5932: 5848: 5784: 5679: 5646: 5544: 5342: 5317: 5242: 5222: 5097: 4976: 4894: 4858: 4723: 4697: 4670: 4624: 4604: 4519: 4451: 4420: 4377: 4353: 4326: 4271: 4250: 4230: 4079: 4056: 4036: 4016: 3996: 3964: 3937: 3917: 3897: 3877: 3660: 3628: 3542: 3489: 3462: 3429: 3383: 3338: 3311: 3273: 2659: 2627: 2577: 2552: 2527: 2502: 2477: 2452: 2424: 2399: 2374: 2349: 2324: 2296: 2271: 2246: 2221: 2193: 2168: 2143: 2115: 2090: 2062: 2056: 1818: 1732: 1706: 1667: 1622: 1583: 1556: 1499: 1479: 1452: 1189: 1169: 1143: 1085: 1038: 991: 827: 709:(1238–1298) defined the triangle, and it is known as 593: 573: 460: 411: 362: 342: 322: 279: 253: 26: 13054:

Fjelstad, P. (1991), "Extending Pascal's Triangle",

10714:{\displaystyle a=\{c-1,-(c+1)\}\;\mathrm {mod} \;2c} 8874: 8719: 7575:

For example, to calculate the diagonal beginning at

4587: +1 ordered partitions form Pascal's triangle. 1805:{\displaystyle (x+1)^{n}=\sum _{k=0}^{n}a_{k}x^{k}.} 11927:

Cambridge University Library: the great collections

9469: 8919: 5523:

The sum of the squares of the elements of row

4658:) is related to the base of the natural logarithm, 2616:

Six rows Pascal's triangle as binomial coefficients

12793: 12766: 12739: 12712: 12685: 12056:Sidoli, Nathan; Brummelen, Glen Van (2013-10-30). 11983: 11728: 11605: 11585: 11553: 11533: 11501: 11471: 11439: 11413: 11386: 11366: 11340: 11320: 11300: 11279: 11228: 11202: 11167: 11028: 11007: 10974: 10896: 10870: 10825: 10780: 10760: 10733: 10713: 10631: 10605: 10573: 10516: 10492: 10472: 10446: 10426: 10384: 10262: 10242: 10212: 10188: 10168: 10137: 10113: 10056: 10031: 9982: 9962: 9919: 9884: 9862: 9775: 9692: 9601: 9408: 9312: 9061: 8904: 8889: 8801: 8708: 8632: 7945: 7923: 7828: 7673: 7608: 7564: 7443: 7396: 7273: 7205: 7137: 7090: 7061: 7032: 7003: 6974: 6942: 6837: 6790: 6689: 6536: 6348: 5959: 5918: 5815: 5770: 5661: 5630: 5506: 5323: 5300: 5228: 5207: 5083: 4844: 4709: 4683: 4637: 4610: 4531: 4505: 4426: 4398: 4359: 4339: 4316:Relation to binomial distribution and convolutions 4304: 4256: 4236: 4212: 4062: 4042: 4022: 4002: 3970: 3950: 3923: 3903: 3883: 3861: 3646: 3611: 3514: 3475: 3448: 3415: 3369: 3324: 3297: 3257: 2645: 2606: 2039: 1804: 1718: 1692: 1653: 1605: 1569: 1540: 1485: 1465: 1438: 1175: 1155: 1118: 1071: 1024: 977: 791:(French: Mr. Pascal's table for combinations) and 611: 579: 559: 444: 389: 348: 328: 291: 265: 201: 12129:"The arithmetical triangle of Jordanus de Nemore" 11271: 11250: 9375:In this triangle, the sum of the elements of row 9050: 9029: 9017: 8988: 8970: 8957: 7532: 7497: 7485: 7464: 6904: 6883: 6871: 6858: 6525: 6498: 5619: 5601: 5583: 5570: 4774: 4761: 4163: 4150: 3812: 3799: 3787: 3766: 3748: 3735: 3723: 3710: 3698: 3685: 3525:It is not difficult to turn this argument into a 2593: 2580: 2568: 2555: 2543: 2530: 2518: 2505: 2493: 2480: 2468: 2455: 2440: 2427: 2415: 2402: 2390: 2377: 2365: 2352: 2340: 2327: 2312: 2299: 2287: 2274: 2262: 2249: 2237: 2224: 2209: 2196: 2184: 2171: 2159: 2146: 2131: 2118: 2106: 2093: 2078: 2065: 1529: 1516: 551: 530: 518: 489: 477: 464: 13332: 10454:can be eliminated from the expansion by setting 9999: 8661:Construction of Clifford algebra using simplices 7989: 7836:, etc. By symmetry, these elements are equal to 7284:To compute the diagonal containing the elements 5332:can be found in Pascal's triangle by use of the 5250: 752:. In Italy, Pascal's triangle is referred to as 12055: 10395:A row corresponds to the binomial expansion of 6047:The diagonals of Pascal's triangle contain the 6043:numbers from a left-justified Pascal's triangle 4598:one right. The sum of the elements of row 4506:{\displaystyle \{\ldots ,0,0,1,1,0,0,\ldots \}} 3958:, as can be seen by observing that each of the 740:(1495–1552) published the full triangle on the 11798:Multiplicities of entries in Pascal's triangle 11016:is formed by concatenating the entries of row 10032:{\displaystyle \lim _{n\to \infty }11_{a}^{n}} 6361:The symmetry of the triangle implies that the 4852:Then, the ratio of successive row products is 4265:entry 3 in row 7 of the above table, which is 3871:In other words, the sum of the entries in the 1661:in terms of the corresponding coefficients of 744:of his book on business calculations in 1527. 638:'s triangle, as depicted by the Chinese using 13159: 13056:Computers & Mathematics with Applications 11974: 11961: 11866:(1949), "The story of the binomial theorem", 10176:) is a univariate polynomial in the variable 7914: 7901: 7887: 7874: 7860: 7847: 7792: 7779: 7738: 7725: 7705: 7692: 7599: 7586: 7428: 7415: 7378: 7357: 7343: 7322: 7308: 7295: 7237: 7224: 7169: 7156: 7122: 7109: 6822: 6809: 6781: 6768: 6748: 6735: 6721: 6708: 5866: 5853: 5761: 5735: 5721: 5703: 5495: 5471: 5462: 5438: 5429: 5405: 4434:increases. This can also be seen by applying 4289: 4276: 1103: 1090: 1056: 1043: 1009: 996: 781:. The triangle was later named for Pascal by 682: 429: 416: 380: 367: 236:, it is named after the French mathematician 13132:Pascal's Treatise on the Arithmetic Triangle 12855:Proceedings of Undergraduate Mathematics Day 12556:An Introduction to Digital Signal Processing 10689: 10656: 4513:with itself corresponds to taking powers of 4500: 4452: 786: 768: 12171:, Oxford University Press, pp. 166–180 12082: 9481:As stated previously, the coefficients of ( 8758:. Unsourced material may be challenged and 796: 13166: 13152: 12884: 12085:Omar Khayyam. The Mathematics Teacher 1958 11984:{\displaystyle \scriptstyle {n \choose k}} 11930:. Cambridge University Press. p. 13. 10704: 10692: 9826:When the factorial function is defined as 9812:, while the general versions are known as 8924:A similar pattern is observed relating to 8854:. Thus, in the tetrahedron, the number of 4555: 13135:(page images of Pascal's treatise, 1654; 13067: 12162: 12160: 12158: 12156: 12144: 11995:is either less than zero or greater than 11923: 10748:By setting the row's radix (the variable 9878: 9436:, there is one vertex at distance 0 from 8778:Learn how and when to remove this message 8702: 4030:at a time, i.e. the number of subsets of 2621:The two summations can be reindexed with 13053: 12326: 12289: 11862: 10878:, respectively. To illustrate, consider 10501: 7949: 7933: 6034: 4570: 4562: 4367:th row of Pascal's triangle becomes the 808: 788:table de M. Pascal pour les combinaisons 653: 630: 307: 12983: 12166: 12107:CRC concise encyclopedia of mathematics 11036:. The twelfth row denotes the product: 9793: 9420:Counting vertices in a cube by distance 6670:Calculating a row or diagonal by itself 6072:The next pair of diagonals contain the 4070:elements, can be found by the equation 13333: 12945: 12911: 12848: 12813: 12662:The Mathematical Works of Isaac Newton 12655: 12617: 12579: 12500: 12367: 12224: 12153: 12126: 11821:, one application of Pascal's triangle 10524:they form the numerical equivalent of 9776:{\displaystyle +i,-1,-i,+1,+i,\ldots } 5527:equals the middle element of row 1163:is raised to a positive integer power 804: 705:(1010–1070). During the 13th century, 13147: 13105: 12867: 12182: 12008: 10040:is the hypothetical terminal row, or 9938: 9821: 6365:d-dimensional number is equal to the 405:". For example, the topmost entry is 13315: 12849:Arnold, Robert; et al. (2004), 12552: 12517: 12440: 12395: 11917: 11858: 11856: 11374:, which is obtained by carrying the 10574:{\displaystyle (a+1)^{n}=11_{a}^{n}} 10114:{\displaystyle (a+1)^{n}=11_{a}^{n}} 8756:adding citations to reliable sources 8723: 6555:The geometric meaning of a function 6016:Every entry in row 2 − 1, 4664:. Specifically, define the sequence 4441:This is related to the operation of 1541:{\displaystyle a_{k}={n \choose k}.} 13018: 12868:Islam, Robiul; et al. (2020), 11541:contains exactly one digit because 9631: 9628: 9540: 9537: 8790:Pascal's triangle can be used as a 7098:, and hence the elements are 5967:can have no prime factors equal to 4410:, this distribution approaches the 4305:{\displaystyle {\tbinom {7}{3}}=35} 3891:th row of Pascal's triangle is the 3483:coefficients in the previous power 798:Triangulum Arithmeticum PASCALIANUM 756:, named for the Italian algebraist 691:. Khayyam used a method of finding 683: 13: 12960:10.1016/B978-0-08-037237-2.50013-1 12632:10.1016/B978-0-08-037237-2.50013-1 12594:10.1016/B978-0-08-037237-2.50013-1 12267:"Pascal's Triangle in Probability" 12127:Hughes, Barnabas (1 August 1989). 11965: 11254: 10871:{\displaystyle 11_{10}^{n}=11^{n}} 10700: 10697: 10694: 10009: 9899: 9842: 9033: 8992: 8961: 8277:Construction as matrix exponential 8123: 7905: 7878: 7851: 7783: 7729: 7696: 7590: 7501: 7468: 7444:{\displaystyle {\tbinom {n}{0}}=1} 7419: 7361: 7326: 7299: 7228: 7160: 7138:{\displaystyle {\tbinom {5}{0}}=1} 7113: 6887: 6862: 6838:{\displaystyle {\tbinom {n}{0}}=1} 6813: 6772: 6739: 6712: 6616:dots composing the target shape. P 6502: 6009:. These numbers are the values in 5857: 5739: 5707: 5605: 5574: 5475: 5442: 5409: 5371: 5260: 4765: 4280: 4154: 3803: 3770: 3739: 3714: 3689: 2584: 2559: 2534: 2509: 2484: 2459: 2431: 2406: 2381: 2356: 2331: 2303: 2278: 2253: 2228: 2200: 2175: 2150: 2122: 2097: 2069: 1726:for simplicity. Suppose then that 1520: 1119:{\displaystyle {\tbinom {2}{2}}=1} 1094: 1072:{\displaystyle {\tbinom {2}{1}}=2} 1047: 1025:{\displaystyle {\tbinom {2}{0}}=1} 1000: 534: 493: 468: 445:{\displaystyle {\tbinom {0}{0}}=1} 420: 371: 14: 13362: 13173: 13080: 12443:The American Mathematical Monthly 12231:The American Mathematical Monthly 12169:Combinatorics: Ancient and Modern 11991:is conventionally set to zero if 11869:The American Mathematical Monthly 11853: 11586:{\displaystyle 1.1_{1234}^{1234}} 8720:Relation to geometry of polytopes 6076:in order, and the next pair give 5091:and the ratio of these ratios is 4399:{\displaystyle p={\tfrac {1}{2}}} 1473:are precisely the numbers in row 775:Treatise on Arithmetical Triangle 336:th row of Pascal's triangle, the 13314: 13305: 13304: 12946:Hilton, P.; et al. (1989). 12618:Hilton, P.; et al. (1989). 12580:Hilton, P.; et al. (1989). 12516:For a similar example, see e.g. 12189:, Cengage Learning, p. 10, 10982:. The numeric representation of 10826:{\displaystyle 11_{1}^{n}=2^{n}} 8920:Number of elements of hypercubes 8728: 8088: 8081: 8067: 8060: 8053: 8044: 8037: 8030: 8023: 8014: 8007: 8000: 7993: 7609:{\displaystyle {\tbinom {5}{0}}} 4105: 4081: 3990:. The number of combinations of 948: 920: 892: 821:. For example, in the expansion 644:Jade Mirror of the Four Unknowns 390:{\displaystyle {\tbinom {n}{k}}} 13047: 13012: 12977: 12939: 12905: 12878: 12861: 12842: 12807: 12649: 12611: 12573: 12546: 12510: 12494: 12485: 12434: 12389: 12361: 12320: 12283: 12259: 12218: 12203: 12176: 12120: 11280:{\displaystyle {n \choose n-1}} 10904:, which yields the row product 9863:{\displaystyle z!=\Gamma (z+1)} 9302: 9298: 9294: 9290: 9286: 9280: 9274: 9259: 9255: 9251: 9247: 9241: 9235: 9220: 9214: 9208: 9202: 9196: 9181: 9175: 9169: 9163: 9148: 9142: 9136: 9121: 9115: 9100: 8802:Number of elements of simplices 7946:Overall patterns and properties 7091:{\displaystyle {\tfrac {1}{5}}} 7062:{\displaystyle {\tfrac {2}{4}}} 7033:{\displaystyle {\tfrac {3}{3}}} 7004:{\displaystyle {\tfrac {4}{2}}} 6975:{\displaystyle {\tfrac {5}{1}}} 3981: 2575: 2550: 2525: 2500: 2475: 2422: 2397: 2372: 2347: 2294: 2269: 2244: 2191: 2166: 2113: 770:Traité du triangle arithmétique 191: 187: 183: 179: 175: 171: 167: 156: 152: 148: 144: 140: 136: 125: 121: 117: 113: 109: 98: 94: 90: 86: 75: 71: 67: 56: 52: 41: 12384:10.5951/mathteacher.108.4.0246 12099: 12076: 12049: 12037: 12002: 11944: 11904: 10686: 10674: 10544: 10531: 10415: 10402: 10084: 10071: 10006: 9914: 9902: 9857: 9845: 9666: 9659: 9575: 9568: 8902:the number of new vertices is 6477: 6471: 6458: 6446: 6398: 6392: 6336: 6324: 6287: 6281: 6231: 6225: 6203: 6191: 6171: 6165: 6139: 6133: 6113: 6107: 5951: 5939: 5906: 5894: 5386: 5376: 5257: 5061: 5048: 4907: 4895: 4833: 4821: 4201: 4189: 4097: 4085: 3834: 3821: 3600: 3578: 3556: 3543: 3503: 3490: 3352: 3339: 3216: 3184: 3098: 3066: 1931: 1918: 1906: 1893: 1878: 1865: 1862: 1850: 1832: 1819: 1746: 1733: 1681: 1668: 1636: 1623: 1203: 1190: 841: 828: 718: 714: 1: 13341:Factorial and binomial topics 12520:Journal of Magnetic Resonance 12399:American Mathematical Monthly 12009:Selin, Helaine (2008-03-12). 11846: 11783:Gaussian binomial coefficient 11749:, Francis Galton's "quincunx" 9798:Pascal's triangle has higher 9785: 3416:{\displaystyle a_{k-1}+a_{k}} 3298:{\displaystyle 0<k<n+1} 612:{\displaystyle 0\leq k\leq n} 13069:10.1016/0898-1221(91)90119-O 12540:10.1016/0022-2364(83)90240-8 12146:10.1016/0315-0860(89)90018-9 11912:History of Indian Literature 10768:) equal to one and ten, row 9920:{\displaystyle \Gamma (z+1)} 9885:{\displaystyle \mathbb {C} } 8709:{\displaystyle \mathbb {R} } 8243: 8233: 8230: 8220: 8216: 8213: 8200: 8184: 8183: 8173: 8170: 8169: 8165: 8162: 8158: 8150: 8145: 6030: 4371:in the symmetric case where 3533:) of the binomial theorem. 1183:, the expression expands as 7: 13093:Encyclopedia of Mathematics 12105:Weisstein, Eric W. (2003). 11739: 11616:represented in radix ten: 11534:{\displaystyle 1.1_{n}^{n}} 9451:and one vertex at distance 8239: 8221: 8204: 8190: 8179: 8146: 5816:{\displaystyle C_{3}=6-1=5} 3370:{\displaystyle (x+1)^{n+1}} 1654:{\displaystyle (x+y)^{n+1}} 10: 13367: 11472:{\displaystyle 11_{n}^{n}} 11008:{\displaystyle 11_{n}^{n}} 8646: 7616:, the fractions are 6585:triangle (a 3-dimensional 783:Pierre Raymond de Montmort 661:'s version of the triangle 626: 303: 13300: 13257: 13226: 13181: 12559:, Elsevier, p. 110, 12347:10.1017/S0025557200004204 11950:The binomial coefficient 11800:(Singmaster's conjecture) 11793:Leibniz harmonic triangle 10500:numeral, as demonstrated 10427:{\displaystyle (a+b)^{n}} 10169:{\displaystyle 14641_{a}} 9948: 9509:of the transform, and if 9470:Fourier transform of sin( 9381:1 + 8 + 24 + 32 + 16 = 81 8872:; the number of faces is 8858:(polyhedral elements) is 3515:{\displaystyle (x+1)^{n}} 3305:, the coefficient of the 1693:{\displaystyle (x+1)^{n}} 646:, a mathematical work by 567:for any positive integer 12334:The Mathematical Gazette 12306:10.4169/math.mag.85.1.51 12256:See in particular p. 11. 11328:. The leftmost digit is 9409:{\displaystyle 3^{4}=81} 9355:-cube, the edge length) 5960:{\displaystyle k!(p-k)!} 5926:. Since the denominator 1606:{\displaystyle x^{n-1}y} 13021:The Mathematics Teacher 12986:The Mathematics Teacher 12816:The Mathematics Teacher 12504:The Algebra Of Geometry 12183:Smith, Karl J. (2010), 11509:. The integral part of 10200:of the variable of the 9970:can be read as a radix 9517:. Then the result is a 8887:the number of edges is 7681:, and the elements are 6020: ≥ 0, is odd. 4710:{\displaystyle n\geq 0} 4591: 4556:Patterns and properties 4224:This is equal to entry 3449:{\displaystyle x^{k-1}} 3332:term in the polynomial 1446:where the coefficients 12912:Kallós, Gábor (2006), 12795: 12794:{\displaystyle 11^{4}} 12768: 12767:{\displaystyle 11^{3}} 12741: 12740:{\displaystyle 11^{2}} 12714: 12713:{\displaystyle 11^{1}} 12687: 12686:{\displaystyle 11^{0}} 12656:Newton, Isaac (1736), 12553:Karl, John H. (2012), 11985: 11730: 11607: 11587: 11555: 11535: 11503: 11473: 11441: 11415: 11414:{\displaystyle 10_{n}} 11388: 11368: 11367:{\displaystyle n>2} 11342: 11322: 11302: 11281: 11230: 11204: 11169: 11030: 11009: 10976: 10898: 10872: 10827: 10782: 10762: 10744:negative row products. 10735: 10715: 10633: 10632:{\displaystyle c<0} 10607: 10575: 10518: 10494: 10474: 10448: 10428: 10386: 10264: 10244: 10214: 10190: 10170: 10139: 10115: 10058: 10033: 9984: 9964: 9921: 9886: 9864: 9777: 9694: 9614:electrical engineering 9603: 9410: 9326:2 = 6 × 2 = 6 × 4 = 24 9314: 9063: 8710: 8634: 8128: 7955: 7942: 7925: 7830: 7675: 7610: 7566: 7445: 7398: 7275: 7207: 7139: 7092: 7063: 7034: 7005: 6976: 6944: 6839: 6792: 6691: 6538: 6445: 6350: 6313: 6264: 6044: 5961: 5920: 5817: 5772: 5663: 5632: 5565: 5536:1 + 4 + 6 + 4 + 1 = 70 5508: 5375: 5325: 5302: 5230: 5209: 5085: 5016: 4952: 4846: 4803: 4757: 4711: 4685: 4639: 4612: 4588: 4568: 4533: 4507: 4428: 4400: 4361: 4341: 4306: 4258: 4238: 4214: 4064: 4044: 4024: 4004: 3972: 3952: 3925: 3905: 3885: 3863: 3681: 3648: 3613: 3531:mathematical induction 3516: 3477: 3450: 3417: 3371: 3326: 3299: 3259: 3183: 3065: 2991: 2889: 2838: 2788: 2734: 2684: 2653:and combined to yield 2647: 2608: 2041: 2013: 1963: 1806: 1778: 1720: 1694: 1655: 1607: 1571: 1542: 1493:of Pascal's triangle: 1487: 1467: 1440: 1235: 1177: 1157: 1120: 1073: 1026: 979: 814: 797: 787: 769: 662: 651: 613: 581: 561: 446: 391: 350: 330: 313: 293: 267: 203: 12921:Annales Mathématiques 12887:Mathematics in School 12796: 12769: 12742: 12715: 12688: 12501:Wilmot, G.P. (2023), 12186:Nature of Mathematics 11986: 11825:Star of David theorem 11788:Hockey-stick identity 11731: 11608: 11588: 11556: 11536: 11504: 11474: 11442: 11416: 11389: 11369: 11343: 11323: 11303: 11282: 11231: 11205: 11203:{\displaystyle k=n-1} 11170: 11031: 11010: 10977: 10899: 10873: 10828: 10783: 10763: 10736: 10716: 10634: 10608: 10606:{\displaystyle c=a+1} 10576: 10519: 10495: 10475: 10449: 10429: 10387: 10265: 10245: 10215: 10191: 10171: 10140: 10116: 10059: 10034: 9985: 9965: 9922: 9887: 9865: 9778: 9695: 9604: 9501:. More precisely: if 9458:(the vertex opposite 9411: 9315: 9064: 8711: 8635: 8127: 7953: 7937: 7926: 7831: 7676: 7611: 7567: 7446: 7399: 7276: 7208: 7140: 7093: 7064: 7035: 7006: 6977: 6945: 6840: 6793: 6692: 6539: 6419: 6372:-dimensional number. 6351: 6293: 6244: 6038: 5962: 5921: 5818: 5773: 5664: 5633: 5545: 5509: 5355: 5326: 5303: 5231: 5210: 5086: 4996: 4926: 4847: 4783: 4737: 4712: 4686: 4684:{\displaystyle s_{n}} 4640: 4638:{\displaystyle 2^{n}} 4613: 4574: 4566: 4534: 4508: 4429: 4408:central limit theorem 4401: 4369:binomial distribution 4362: 4342: 4340:{\displaystyle 2^{n}} 4307: 4259: 4239: 4215: 4065: 4045: 4025: 4005: 3973: 3953: 3951:{\displaystyle 2^{n}} 3926: 3906: 3886: 3864: 3661: 3649: 3647:{\displaystyle x=y=1} 3614: 3517: 3478: 3476:{\displaystyle x^{k}} 3451: 3418: 3372: 3327: 3325:{\displaystyle x^{k}} 3300: 3260: 3163: 3045: 2971: 2869: 2818: 2762: 2714: 2664: 2648: 2646:{\displaystyle k=i+1} 2609: 2042: 1993: 1943: 1807: 1758: 1721: 1695: 1656: 1608: 1572: 1570:{\displaystyle x^{n}} 1543: 1488: 1468: 1466:{\displaystyle a_{k}} 1441: 1215: 1178: 1158: 1121: 1074: 1027: 980: 812: 795:(1730) who called it 785:(1708) who called it 671:binomial coefficients 657: 634: 614: 582: 562: 447: 392: 351: 331: 311: 294: 268: 230:binomial coefficients 204: 13351:Triangles of numbers 13235:Lettres provinciales 13033:10.5951/MT.59.5.0461 12998:10.5951/MT.58.5.0425 12828:10.5951/MT.57.6.0392 12778: 12751: 12724: 12697: 12670: 12293:Mathematics Magazine 12133:Historia Mathematica 12083:Kennedy, E. (1966). 11954: 11910:Maurice Winternitz, 11758:Bernoulli's triangle 11623: 11606:{\displaystyle 1234} 11597: 11565: 11545: 11513: 11487: 11451: 11425: 11398: 11378: 11352: 11332: 11312: 11292: 11244: 11214: 11182: 11043: 11020: 10987: 10908: 10882: 10837: 10792: 10788:becomes the product 10772: 10752: 10725: 10647: 10617: 10585: 10528: 10508: 10484: 10458: 10438: 10399: 10274: 10254: 10228: 10204: 10180: 10153: 10129: 10068: 10048: 9995: 9974: 9954: 9896: 9874: 9830: 9810:Pascal's tetrahedron 9794:To higher dimensions 9722: 9623: 9532: 9387: 9383:, which is equal to 9079: 8951: 8752:improve this section 8698: 8286: 7941:in Pascal's triangle 7840: 7685: 7620: 7579: 7458: 7408: 7288: 7217: 7149: 7102: 7073: 7044: 7015: 6986: 6957: 6852: 6802: 6701: 6681: 6379: 6090: 5930: 5846: 5782: 5677: 5662:{\displaystyle n=2m} 5644: 5542: 5340: 5324:{\displaystyle \pi } 5315: 5240: 5220: 5095: 4856: 4721: 4695: 4668: 4622: 4602: 4517: 4449: 4443:discrete convolution 4418: 4375: 4351: 4324: 4269: 4248: 4228: 4077: 4054: 4050:elements from among 4034: 4014: 3994: 3962: 3935: 3915: 3895: 3875: 3658: 3626: 3540: 3487: 3460: 3427: 3381: 3336: 3309: 3271: 2657: 2625: 2054: 1816: 1730: 1704: 1665: 1620: 1581: 1554: 1497: 1477: 1450: 1187: 1167: 1141: 1083: 1036: 989: 825: 754:Tartaglia's triangle 591: 571: 458: 409: 360: 356:th entry is denoted 340: 320: 277: 251: 24: 13198:Pascal's calculator 13110:"Pascal's triangle" 12532:1983JMagR..55..283H 12480:Théorie des nombres 12371:Mathematics Teacher 12271:5010.mathed.usu.edu 12021:2008ehst.book.....S 11830:Trinomial expansion 11640: 11582: 11530: 11502:{\displaystyle n+1} 11483:to the row length, 11468: 11440:{\displaystyle k=1} 11229:{\displaystyle k=1} 11060: 11004: 10971: 10897:{\displaystyle a=n} 10854: 10809: 10721:with odd values of 10639:, then the theorem 10570: 10473:{\displaystyle b=1} 10243:{\displaystyle i=0} 10147:positional notation 10110: 10028: 9466:in each dimension. 7965:Sierpinski triangle 6612:), have a total of 6074:tetrahedral numbers 5538:. In general form, 5151: 4532:{\displaystyle x+1} 4412:normal distribution 4119: 1719:{\displaystyle y=1} 1156:{\displaystyle x+y} 1133:states that when a 819:binomial expansions 805:Binomial expansions 711:Yang Hui's triangle 292:{\displaystyle k=0} 266:{\displaystyle n=0} 13107:Weisstein, Eric W. 12791: 12764: 12737: 12710: 12683: 11981: 11980: 11924:Peter Fox (1998). 11835:Trinomial triangle 11763:Binomial expansion 11726: 11626: 11603: 11583: 11568: 11551: 11531: 11516: 11499: 11469: 11454: 11437: 11411: 11384: 11364: 11338: 11318: 11301:{\displaystyle 01} 11298: 11277: 11226: 11200: 11165: 11046: 11026: 11005: 10990: 10972: 10957: 10894: 10868: 10840: 10823: 10795: 10778: 10758: 10731: 10711: 10629: 10603: 10571: 10556: 10514: 10490: 10470: 10444: 10424: 10382: 10260: 10240: 10210: 10186: 10166: 10135: 10111: 10096: 10054: 10029: 10014: 10013: 9980: 9960: 9939:To arbitrary bases 9917: 9882: 9860: 9822:To complex numbers 9815:Pascal's simplices 9773: 9690: 9599: 9505:is even, take the 9432:: fixing a vertex 9406: 9310: 9308: 9059: 8706: 8655:matrix exponential 8630: 8628: 8594: 8440: 8129: 7984:tends to infinity. 7956: 7943: 7939:Fibonacci sequence 7921: 7919: 7892: 7865: 7826: 7818: 7797: 7764: 7743: 7710: 7671: 7663: 7648: 7633: 7606: 7604: 7562: 7441: 7433: 7394: 7383: 7348: 7313: 7271: 7263: 7242: 7203: 7195: 7174: 7135: 7127: 7088: 7086: 7059: 7057: 7030: 7028: 7001: 6999: 6972: 6970: 6940: 6835: 6827: 6788: 6786: 6753: 6726: 6697:with the elements 6687: 6534: 6346: 6344: 6067:triangular numbers 6045: 5989:terms in row 5957: 5916: 5914: 5871: 5813: 5768: 5766: 5726: 5659: 5628: 5518:Lozanić's triangle 5504: 5321: 5298: 5264: 5226: 5205: 5137: 5081: 5039: 4975: 4842: 4707: 4681: 4635: 4608: 4589: 4569: 4529: 4503: 4436:Stirling's formula 4424: 4396: 4394: 4357: 4337: 4302: 4294: 4254: 4234: 4210: 4103: 4060: 4040: 4020: 4000: 3968: 3948: 3921: 3901: 3881: 3859: 3644: 3609: 3591: 3512: 3473: 3446: 3413: 3367: 3322: 3295: 3255: 3253: 2643: 2604: 2602: 2598: 2573: 2548: 2523: 2498: 2473: 2445: 2420: 2395: 2370: 2345: 2317: 2292: 2267: 2242: 2214: 2189: 2164: 2136: 2111: 2083: 2037: 1802: 1716: 1690: 1651: 1603: 1567: 1538: 1483: 1463: 1436: 1173: 1153: 1116: 1108: 1069: 1061: 1022: 1014: 975: 815: 779:probability theory 730:Jordanus de Nemore 679:Khayyam's triangle 663: 652: 609: 577: 557: 442: 434: 387: 385: 346: 326: 314: 289: 263: 199: 197: 13328: 13327: 13289:Marguerite Périer 13273:Jacqueline Pascal 13239:(1656–1657) 13213:Pascal's triangle 13088:"Pascal triangle" 12115:978-1-58488-347-0 11972: 11937:978-0-521-62647-7 11768:Cellular automata 11690: 11688: 11683: 11554:{\displaystyle n} 11387:{\displaystyle 1} 11341:{\displaystyle 2} 11321:{\displaystyle n} 11269: 11029:{\displaystyle n} 10941: 10781:{\displaystyle n} 10761:{\displaystyle a} 10734:{\displaystyle n} 10517:{\displaystyle a} 10493:{\displaystyle a} 10447:{\displaystyle b} 10263:{\displaystyle i} 10213:{\displaystyle i} 10189:{\displaystyle a} 10138:{\displaystyle a} 10057:{\displaystyle n} 9998: 9983:{\displaystyle a} 9963:{\displaystyle n} 9679: 9644: 9588: 9553: 9513:is odd, take the 9491:Fourier transform 9284: 9278: 9272: 9263: 9245: 9239: 9233: 9224: 9218: 9212: 9206: 9200: 9194: 9185: 9179: 9173: 9167: 9161: 9152: 9146: 9140: 9134: 9125: 9119: 9113: 9104: 9098: 9089: 9048: 9015: 8968: 8827:-simplex from an 8788: 8787: 8780: 8687:Geometric algebra 8671:Geometric Algebra 8669:used as forms in 8624: 8611: 8270: 8269: 8134:Fibonacci numbers 8106: 8105: 7912: 7885: 7858: 7817: 7790: 7763: 7736: 7703: 7662: 7647: 7632: 7597: 7557: 7530: 7483: 7426: 7404:begin again with 7376: 7341: 7306: 7262: 7235: 7194: 7167: 7120: 7085: 7056: 7027: 6998: 6969: 6935: 6902: 6869: 6820: 6779: 6746: 6719: 6690:{\displaystyle n} 6523: 6490: 6417: 6078:pentatope numbers 5913: 5864: 5759: 5719: 5617: 5581: 5502: 5493: 5460: 5427: 5336:infinite series. 5285: 5249: 5229:{\displaystyle e} 5195: 5178: 5152: 5079: 5040: 5037: 4973: 4887: 4840: 4772: 4611:{\displaystyle n} 4550:convolution power 4427:{\displaystyle n} 4393: 4360:{\displaystyle n} 4287: 4257:{\displaystyle n} 4237:{\displaystyle k} 4208: 4161: 4063:{\displaystyle n} 4043:{\displaystyle k} 4023:{\displaystyle k} 4003:{\displaystyle n} 3971:{\displaystyle n} 3924:{\displaystyle n} 3904:{\displaystyle n} 3884:{\displaystyle n} 3810: 3785: 3746: 3721: 3696: 3590: 3423:, the sum of the 2591: 2566: 2541: 2516: 2491: 2466: 2438: 2413: 2388: 2363: 2338: 2310: 2285: 2260: 2235: 2207: 2182: 2157: 2129: 2104: 2076: 1527: 1486:{\displaystyle n} 1176:{\displaystyle n} 1101: 1054: 1007: 793:Abraham de Moivre 580:{\displaystyle n} 549: 516: 475: 427: 378: 349:{\displaystyle k} 329:{\displaystyle n} 240:, although other 222:Pascal's triangle 13358: 13318: 13317: 13308: 13307: 13293: 13285: 13277: 13269: 13250: 13240: 13208:Pascal's theorem 13168: 13161: 13154: 13145: 13144: 13120: 13119: 13101: 13074: 13072: 13071: 13051: 13045: 13043: 13016: 13010: 13008: 12981: 12975: 12973: 12943: 12937: 12935: 12933:10.5802/ambp.211 12918: 12909: 12903: 12901: 12882: 12876: 12874: 12865: 12859: 12857: 12846: 12840: 12838: 12811: 12805: 12803: 12800: 12798: 12797: 12792: 12790: 12789: 12773: 12771: 12770: 12765: 12763: 12762: 12746: 12744: 12743: 12738: 12736: 12735: 12719: 12717: 12716: 12711: 12709: 12708: 12692: 12690: 12689: 12684: 12682: 12681: 12653: 12647: 12645: 12615: 12609: 12607: 12577: 12571: 12569: 12550: 12544: 12542: 12514: 12508: 12507: 12498: 12492: 12489: 12483: 12473: 12438: 12432: 12430: 12393: 12387: 12386: 12365: 12359: 12357: 12324: 12318: 12316: 12287: 12281: 12280: 12278: 12277: 12263: 12257: 12255: 12222: 12216: 12207: 12201: 12199: 12180: 12174: 12172: 12164: 12151: 12150: 12148: 12124: 12118: 12103: 12097: 12096: 12080: 12074: 12073: 12053: 12047: 12041: 12035: 12034: 12006: 12000: 11990: 11988: 11987: 11982: 11979: 11978: 11977: 11964: 11948: 11942: 11941: 11921: 11915: 11908: 11902: 11900: 11860: 11814:Pascal's simplex 11809:Pascal's pyramid 11778:Floyd's triangle 11735: 11733: 11732: 11727: 11725: 11724: 11709: 11708: 11689: 11686: 11684: 11676: 11674: 11639: 11634: 11612: 11610: 11609: 11604: 11592: 11590: 11589: 11584: 11581: 11576: 11560: 11558: 11557: 11552: 11540: 11538: 11537: 11532: 11529: 11524: 11508: 11506: 11505: 11500: 11478: 11476: 11475: 11470: 11467: 11462: 11446: 11444: 11443: 11438: 11420: 11418: 11417: 11412: 11410: 11409: 11393: 11391: 11390: 11385: 11373: 11371: 11370: 11365: 11347: 11345: 11344: 11339: 11327: 11325: 11324: 11319: 11307: 11305: 11304: 11299: 11286: 11284: 11283: 11278: 11276: 11275: 11274: 11268: 11253: 11235: 11233: 11232: 11227: 11209: 11207: 11206: 11201: 11174: 11172: 11171: 11166: 11164: 11163: 11145: 11144: 11059: 11054: 11035: 11033: 11032: 11027: 11014: 11012: 11011: 11006: 11003: 10998: 10981: 10979: 10978: 10973: 10970: 10965: 10953: 10952: 10947: 10943: 10942: 10934: 10920: 10919: 10903: 10901: 10900: 10895: 10877: 10875: 10874: 10869: 10867: 10866: 10853: 10848: 10832: 10830: 10829: 10824: 10822: 10821: 10808: 10803: 10787: 10785: 10784: 10779: 10767: 10765: 10764: 10759: 10740: 10738: 10737: 10732: 10720: 10718: 10717: 10712: 10703: 10638: 10636: 10635: 10630: 10612: 10610: 10609: 10604: 10580: 10578: 10577: 10572: 10569: 10564: 10552: 10551: 10523: 10521: 10520: 10515: 10499: 10497: 10496: 10491: 10479: 10477: 10476: 10471: 10453: 10451: 10450: 10445: 10433: 10431: 10430: 10425: 10423: 10422: 10391: 10389: 10388: 10383: 10381: 10380: 10362: 10361: 10343: 10342: 10324: 10323: 10305: 10304: 10286: 10285: 10269: 10267: 10266: 10261: 10249: 10247: 10246: 10241: 10219: 10217: 10216: 10211: 10195: 10193: 10192: 10187: 10175: 10173: 10172: 10167: 10165: 10164: 10144: 10142: 10141: 10136: 10120: 10118: 10117: 10112: 10109: 10104: 10092: 10091: 10063: 10061: 10060: 10055: 10038: 10036: 10035: 10030: 10027: 10022: 10012: 9989: 9987: 9986: 9981: 9969: 9967: 9966: 9961: 9926: 9924: 9923: 9918: 9891: 9889: 9888: 9883: 9881: 9869: 9867: 9866: 9861: 9805:Pascal's pyramid 9782: 9780: 9779: 9774: 9699: 9697: 9696: 9691: 9689: 9685: 9684: 9680: 9675: 9674: 9673: 9651: 9645: 9642: 9635: 9634: 9608: 9606: 9605: 9600: 9598: 9594: 9593: 9589: 9584: 9583: 9582: 9560: 9554: 9551: 9544: 9543: 9457: 9456: 9450: 9449: 9415: 9413: 9412: 9407: 9399: 9398: 9382: 9350: 9327: 9319: 9317: 9316: 9311: 9309: 9285: 9282: 9279: 9276: 9273: 9270: 9264: 9261: 9246: 9243: 9240: 9237: 9234: 9231: 9225: 9222: 9219: 9216: 9213: 9210: 9207: 9204: 9201: 9198: 9195: 9192: 9186: 9183: 9180: 9177: 9174: 9171: 9168: 9165: 9162: 9159: 9153: 9150: 9147: 9144: 9141: 9138: 9135: 9132: 9126: 9123: 9120: 9117: 9114: 9111: 9105: 9102: 9099: 9096: 9090: 9087: 9068: 9066: 9065: 9060: 9055: 9054: 9053: 9044: 9032: 9022: 9021: 9020: 9014: 9003: 8991: 8975: 8974: 8973: 8960: 8943: 8935: 8915: 8910: 8906: 8901: 8895: 8891: 8886: 8880: 8876: 8871: 8866: 8862: 8834: 8783: 8776: 8772: 8769: 8763: 8732: 8724: 8716:, with basis 1. 8715: 8713: 8712: 8707: 8705: 8691:binomial theorem 8684: 8680: 8676: 8667:Clifford algebra 8639: 8637: 8636: 8631: 8629: 8625: 8622: 8613: 8612: 8609: 8599: 8598: 8445: 8444: 8144: 8143: 8102: 8097: 8092: 8085: 8076: 8071: 8064: 8057: 8048: 8041: 8034: 8027: 8018: 8011: 8004: 7997: 7990: 7930: 7928: 7927: 7922: 7920: 7918: 7917: 7904: 7893: 7891: 7890: 7877: 7866: 7864: 7863: 7850: 7835: 7833: 7832: 7827: 7819: 7810: 7798: 7796: 7795: 7782: 7765: 7756: 7744: 7742: 7741: 7728: 7711: 7709: 7708: 7695: 7680: 7678: 7677: 7672: 7664: 7655: 7649: 7640: 7634: 7625: 7615: 7613: 7612: 7607: 7605: 7603: 7602: 7589: 7571: 7569: 7568: 7563: 7558: 7553: 7542: 7537: 7536: 7535: 7529: 7518: 7500: 7490: 7489: 7488: 7479: 7467: 7450: 7448: 7447: 7442: 7434: 7432: 7431: 7418: 7403: 7401: 7400: 7395: 7384: 7382: 7381: 7372: 7360: 7349: 7347: 7346: 7337: 7325: 7314: 7312: 7311: 7298: 7280: 7278: 7277: 7272: 7264: 7255: 7243: 7241: 7240: 7227: 7212: 7210: 7209: 7204: 7196: 7187: 7175: 7173: 7172: 7159: 7144: 7142: 7141: 7136: 7128: 7126: 7125: 7112: 7097: 7095: 7094: 7089: 7087: 7078: 7068: 7066: 7065: 7060: 7058: 7049: 7039: 7037: 7036: 7031: 7029: 7020: 7010: 7008: 7007: 7002: 7000: 6991: 6981: 6979: 6978: 6973: 6971: 6962: 6949: 6947: 6946: 6941: 6936: 6931: 6914: 6909: 6908: 6907: 6901: 6886: 6876: 6875: 6874: 6861: 6844: 6842: 6841: 6836: 6828: 6826: 6825: 6812: 6797: 6795: 6794: 6789: 6787: 6785: 6784: 6771: 6754: 6752: 6751: 6738: 6727: 6725: 6724: 6711: 6696: 6694: 6693: 6688: 6573:(1) = 1 for all 6550:rising factorial 6543: 6541: 6540: 6535: 6530: 6529: 6528: 6519: 6501: 6491: 6489: 6481: 6480: 6465: 6444: 6433: 6418: 6416: 6405: 6391: 6390: 6355: 6353: 6352: 6347: 6345: 6323: 6322: 6312: 6307: 6280: 6279: 6263: 6258: 6237: 6224: 6223: 6190: 6189: 6164: 6163: 6132: 6131: 6106: 6105: 6049:figurate numbers 6011:Gould's sequence 6008: 6004: 5996: 5992: 5978: 5974: 5970: 5966: 5964: 5963: 5958: 5925: 5923: 5922: 5917: 5915: 5912: 5886: 5878: 5872: 5870: 5869: 5856: 5841: 5833: 5829: 5822: 5820: 5819: 5814: 5794: 5793: 5777: 5775: 5774: 5769: 5767: 5765: 5764: 5758: 5747: 5738: 5727: 5725: 5724: 5715: 5706: 5695: 5694: 5668: 5666: 5665: 5660: 5640:In any even row 5637: 5635: 5634: 5629: 5624: 5623: 5622: 5613: 5604: 5594: 5593: 5588: 5587: 5586: 5573: 5564: 5559: 5537: 5533: 5526: 5513: 5511: 5510: 5505: 5503: 5501: 5500: 5499: 5498: 5489: 5474: 5467: 5466: 5465: 5456: 5441: 5433: 5432: 5423: 5408: 5402: 5400: 5399: 5374: 5369: 5330: 5328: 5327: 5322: 5307: 5305: 5304: 5299: 5297: 5296: 5291: 5287: 5286: 5278: 5263: 5235: 5233: 5232: 5227: 5214: 5212: 5211: 5206: 5193: 5189: 5188: 5183: 5179: 5174: 5163: 5153: 5150: 5145: 5136: 5135: 5134: 5116: 5115: 5099: 5090: 5088: 5087: 5082: 5080: 5078: 5070: 5069: 5068: 5046: 5041: 5038: 5036: 5035: 5034: 5018: 5015: 5010: 4995: 4994: 4974: 4972: 4971: 4970: 4954: 4951: 4940: 4925: 4924: 4893: 4888: 4886: 4885: 4876: 4875: 4860: 4851: 4849: 4848: 4843: 4841: 4839: 4813: 4805: 4802: 4797: 4779: 4778: 4777: 4764: 4756: 4751: 4733: 4732: 4716: 4714: 4713: 4708: 4690: 4688: 4687: 4682: 4680: 4679: 4653: 4644: 4642: 4641: 4636: 4634: 4633: 4617: 4615: 4614: 4609: 4538: 4536: 4535: 4530: 4512: 4510: 4509: 4504: 4433: 4431: 4430: 4425: 4405: 4403: 4402: 4397: 4395: 4386: 4366: 4364: 4363: 4358: 4346: 4344: 4343: 4338: 4336: 4335: 4320:When divided by 4311: 4309: 4308: 4303: 4295: 4293: 4292: 4279: 4263: 4261: 4260: 4255: 4243: 4241: 4240: 4235: 4219: 4217: 4216: 4211: 4209: 4207: 4181: 4173: 4168: 4167: 4166: 4153: 4143: 4142: 4141: 4132: 4131: 4118: 4113: 4108: 4084: 4069: 4067: 4066: 4061: 4049: 4047: 4046: 4041: 4029: 4027: 4026: 4021: 4009: 4007: 4006: 4001: 3977: 3975: 3974: 3969: 3957: 3955: 3954: 3949: 3947: 3946: 3931:-element set is 3930: 3928: 3927: 3922: 3910: 3908: 3907: 3902: 3890: 3888: 3887: 3882: 3868: 3866: 3865: 3860: 3855: 3854: 3842: 3841: 3817: 3816: 3815: 3802: 3792: 3791: 3790: 3784: 3769: 3753: 3752: 3751: 3738: 3728: 3727: 3726: 3713: 3703: 3702: 3701: 3688: 3680: 3675: 3653: 3651: 3650: 3645: 3618: 3616: 3615: 3610: 3608: 3607: 3592: 3583: 3577: 3576: 3564: 3563: 3521: 3519: 3518: 3513: 3511: 3510: 3482: 3480: 3479: 3474: 3472: 3471: 3455: 3453: 3452: 3447: 3445: 3444: 3422: 3420: 3419: 3414: 3412: 3411: 3399: 3398: 3376: 3374: 3373: 3368: 3366: 3365: 3331: 3329: 3328: 3323: 3321: 3320: 3304: 3302: 3301: 3296: 3264: 3262: 3261: 3256: 3254: 3247: 3246: 3228: 3227: 3215: 3214: 3202: 3201: 3182: 3177: 3159: 3158: 3143: 3139: 3138: 3123: 3122: 3110: 3109: 3097: 3096: 3084: 3083: 3064: 3059: 3041: 3040: 3031: 3030: 3015: 3011: 3010: 3001: 3000: 2990: 2985: 2967: 2966: 2957: 2956: 2944: 2943: 2928: 2927: 2915: 2914: 2905: 2904: 2888: 2883: 2862: 2858: 2857: 2848: 2847: 2837: 2832: 2814: 2813: 2804: 2803: 2787: 2776: 2754: 2753: 2744: 2743: 2733: 2728: 2710: 2709: 2694: 2693: 2683: 2678: 2652: 2650: 2649: 2644: 2613: 2611: 2610: 2605: 2603: 2599: 2597: 2596: 2583: 2574: 2572: 2571: 2558: 2549: 2547: 2546: 2533: 2524: 2522: 2521: 2508: 2499: 2497: 2496: 2483: 2474: 2472: 2471: 2458: 2446: 2444: 2443: 2430: 2421: 2419: 2418: 2405: 2396: 2394: 2393: 2380: 2371: 2369: 2368: 2355: 2346: 2344: 2343: 2330: 2318: 2316: 2315: 2302: 2293: 2291: 2290: 2277: 2268: 2266: 2265: 2252: 2243: 2241: 2240: 2227: 2215: 2213: 2212: 2199: 2190: 2188: 2187: 2174: 2165: 2163: 2162: 2149: 2137: 2135: 2134: 2121: 2112: 2110: 2109: 2096: 2084: 2082: 2081: 2068: 2046: 2044: 2043: 2038: 2033: 2032: 2023: 2022: 2012: 2007: 1989: 1988: 1973: 1972: 1962: 1957: 1939: 1938: 1914: 1913: 1886: 1885: 1846: 1845: 1811: 1809: 1808: 1803: 1798: 1797: 1788: 1787: 1777: 1772: 1754: 1753: 1725: 1723: 1722: 1717: 1700:, where we sett 1699: 1697: 1696: 1691: 1689: 1688: 1660: 1658: 1657: 1652: 1650: 1649: 1612: 1610: 1609: 1604: 1599: 1598: 1576: 1574: 1573: 1568: 1566: 1565: 1547: 1545: 1544: 1539: 1534: 1533: 1532: 1519: 1509: 1508: 1492: 1490: 1489: 1484: 1472: 1470: 1469: 1464: 1462: 1461: 1445: 1443: 1442: 1437: 1432: 1431: 1422: 1421: 1409: 1408: 1390: 1389: 1365: 1364: 1355: 1354: 1339: 1338: 1323: 1322: 1307: 1306: 1294: 1293: 1284: 1283: 1271: 1270: 1261: 1260: 1245: 1244: 1234: 1229: 1211: 1210: 1182: 1180: 1179: 1174: 1162: 1160: 1159: 1154: 1131:binomial theorem 1129:In general, the 1125: 1123: 1122: 1117: 1109: 1107: 1106: 1093: 1078: 1076: 1075: 1070: 1062: 1060: 1059: 1046: 1031: 1029: 1028: 1023: 1015: 1013: 1012: 999: 984: 982: 981: 976: 971: 970: 961: 960: 951: 943: 942: 933: 932: 923: 915: 914: 905: 904: 895: 887: 886: 862: 861: 849: 848: 800: 790: 772: 762:Gerolamo Cardano 750:figurate numbers 720: 716: 689:binomial theorem 686: 685: 618: 616: 615: 610: 587:and any integer 586: 584: 583: 578: 566: 564: 563: 558: 556: 555: 554: 545: 533: 523: 522: 521: 515: 504: 492: 482: 481: 480: 467: 451: 449: 448: 443: 435: 433: 432: 419: 404: 400: 396: 394: 393: 388: 386: 384: 383: 370: 355: 353: 352: 347: 335: 333: 332: 327: 298: 296: 295: 290: 272: 270: 269: 264: 226:triangular array 208: 206: 205: 200: 198: 13366: 13365: 13361: 13360: 13359: 13357: 13356: 13355: 13331: 13330: 13329: 13324: 13296: 13291: 13283: 13281:Gilberte Périer 13275: 13267: 13253: 13248: 13238: 13222: 13191: 13177: 13172: 13086: 13083: 13078: 13077: 13052: 13048: 13017: 13013: 12982: 12978: 12970: 12944: 12940: 12916: 12910: 12906: 12883: 12879: 12866: 12862: 12847: 12843: 12812: 12808: 12785: 12781: 12779: 12776: 12775: 12758: 12754: 12752: 12749: 12748: 12731: 12727: 12725: 12722: 12721: 12704: 12700: 12698: 12695: 12694: 12677: 12673: 12671: 12668: 12667: 12654: 12650: 12642: 12616: 12612: 12604: 12578: 12574: 12567: 12551: 12547: 12515: 12511: 12499: 12495: 12490: 12486: 12455:10.2307/2324061 12439: 12435: 12412:10.2307/2304500 12406:(10): 589–592, 12394: 12390: 12366: 12362: 12325: 12321: 12288: 12284: 12275: 12273: 12265: 12264: 12260: 12244:10.2307/2975209 12223: 12219: 12208: 12204: 12197: 12181: 12177: 12165: 12154: 12125: 12121: 12104: 12100: 12081: 12077: 12070: 12054: 12050: 12042: 12038: 12031: 12007: 12003: 11973: 11960: 11959: 11958: 11955: 11952: 11951: 11949: 11945: 11938: 11922: 11918: 11909: 11905: 11882:10.2307/2305028 11864:Coolidge, J. L. 11861: 11854: 11849: 11844: 11742: 11720: 11716: 11704: 11700: 11685: 11675: 11635: 11630: 11624: 11621: 11620: 11598: 11595: 11594: 11577: 11572: 11566: 11563: 11562: 11546: 11543: 11542: 11525: 11520: 11514: 11511: 11510: 11488: 11485: 11484: 11463: 11458: 11452: 11449: 11448: 11426: 11423: 11422: 11405: 11401: 11399: 11396: 11395: 11379: 11376: 11375: 11353: 11350: 11349: 11333: 11330: 11329: 11313: 11310: 11309: 11293: 11290: 11289: 11270: 11258: 11249: 11248: 11247: 11245: 11242: 11241: 11215: 11212: 11211: 11183: 11180: 11179: 11159: 11155: 11140: 11136: 11055: 11050: 11044: 11041: 11040: 11021: 11018: 11017: 10999: 10994: 10988: 10985: 10984: 10966: 10961: 10948: 10933: 10926: 10922: 10921: 10915: 10911: 10909: 10906: 10905: 10883: 10880: 10879: 10862: 10858: 10849: 10844: 10838: 10835: 10834: 10817: 10813: 10804: 10799: 10793: 10790: 10789: 10773: 10770: 10769: 10753: 10750: 10749: 10726: 10723: 10722: 10693: 10648: 10645: 10644: 10618: 10615: 10614: 10586: 10583: 10582: 10565: 10560: 10547: 10543: 10529: 10526: 10525: 10509: 10506: 10505: 10485: 10482: 10481: 10459: 10456: 10455: 10439: 10436: 10435: 10434:. The variable 10418: 10414: 10400: 10397: 10396: 10376: 10372: 10357: 10353: 10338: 10334: 10319: 10315: 10300: 10296: 10281: 10277: 10275: 10272: 10271: 10270:. For example, 10255: 10252: 10251: 10229: 10226: 10225: 10224:(starting with 10205: 10202: 10201: 10181: 10178: 10177: 10160: 10156: 10154: 10151: 10150: 10130: 10127: 10126: 10105: 10100: 10087: 10083: 10069: 10066: 10065: 10049: 10046: 10045: 10023: 10018: 10002: 9996: 9993: 9992: 9990:numeral, where 9975: 9972: 9971: 9955: 9952: 9951: 9941: 9897: 9894: 9893: 9877: 9875: 9872: 9871: 9831: 9828: 9827: 9824: 9796: 9788: 9723: 9720: 9719: 9705:boxcar function 9669: 9665: 9652: 9650: 9646: 9641: 9640: 9636: 9627: 9626: 9624: 9621: 9620: 9578: 9574: 9561: 9559: 9555: 9550: 9549: 9545: 9536: 9535: 9533: 9530: 9529: 9479: 9454: 9452: 9447: 9445: 9422: 9394: 9390: 9388: 9385: 9384: 9380: 9344: 9325: 9307: 9306: 9281: 9275: 9269: 9266: 9265: 9260: 9242: 9236: 9230: 9227: 9226: 9221: 9215: 9209: 9203: 9197: 9191: 9188: 9187: 9182: 9176: 9170: 9164: 9158: 9155: 9154: 9149: 9143: 9137: 9131: 9128: 9127: 9122: 9116: 9110: 9107: 9106: 9101: 9095: 9092: 9091: 9086: 9082: 9080: 9077: 9076: 9049: 9034: 9028: 9027: 9026: 9016: 9004: 8993: 8987: 8986: 8985: 8969: 8956: 8955: 8954: 8952: 8949: 8948: 8937: 8929: 8922: 8903: 8888: 8873: 8859: 8828: 8804: 8784: 8773: 8767: 8764: 8749: 8733: 8722: 8701: 8699: 8696: 8695: 8682: 8678: 8674: 8663: 8651: 8645: 8644: 8643: 8640: 8627: 8626: 8621: 8614: 8608: 8604: 8601: 8600: 8593: 8592: 8587: 8582: 8577: 8572: 8566: 8565: 8560: 8555: 8550: 8545: 8539: 8538: 8533: 8528: 8523: 8518: 8512: 8511: 8506: 8501: 8496: 8491: 8485: 8484: 8479: 8474: 8469: 8464: 8454: 8453: 8446: 8439: 8438: 8433: 8428: 8423: 8418: 8412: 8411: 8406: 8401: 8396: 8391: 8385: 8384: 8379: 8374: 8369: 8364: 8358: 8357: 8352: 8347: 8342: 8337: 8331: 8330: 8325: 8320: 8315: 8310: 8300: 8299: 8289: 8287: 8284: 8283: 8279: 8113: 8112: 8111: 8100: 8095: 8074: 7948: 7913: 7900: 7899: 7897: 7886: 7873: 7872: 7870: 7859: 7846: 7845: 7843: 7841: 7838: 7837: 7808: 7791: 7778: 7777: 7775: 7754: 7737: 7724: 7723: 7721: 7704: 7691: 7690: 7688: 7686: 7683: 7682: 7653: 7638: 7623: 7621: 7618: 7617: 7598: 7585: 7584: 7582: 7580: 7577: 7576: 7543: 7541: 7531: 7519: 7502: 7496: 7495: 7494: 7484: 7469: 7463: 7462: 7461: 7459: 7456: 7455: 7427: 7414: 7413: 7411: 7409: 7406: 7405: 7377: 7362: 7356: 7355: 7353: 7342: 7327: 7321: 7320: 7318: 7307: 7294: 7293: 7291: 7289: 7286: 7285: 7253: 7236: 7223: 7222: 7220: 7218: 7215: 7214: 7185: 7168: 7155: 7154: 7152: 7150: 7147: 7146: 7121: 7108: 7107: 7105: 7103: 7100: 7099: 7076: 7074: 7071: 7070: 7047: 7045: 7042: 7041: 7018: 7016: 7013: 7012: 6989: 6987: 6984: 6983: 6960: 6958: 6955: 6954: 6915: 6913: 6903: 6891: 6882: 6881: 6880: 6870: 6857: 6856: 6855: 6853: 6850: 6849: 6821: 6808: 6807: 6805: 6803: 6800: 6799: 6780: 6767: 6766: 6764: 6747: 6734: 6733: 6731: 6720: 6707: 6706: 6704: 6702: 6699: 6698: 6682: 6679: 6678: 6677:To compute row 6672: 6661: 6643: 6632: 6621: 6607: 6601: 6572: 6563: 6524: 6503: 6497: 6496: 6495: 6482: 6470: 6466: 6464: 6434: 6423: 6409: 6404: 6386: 6382: 6380: 6377: 6376: 6343: 6342: 6318: 6314: 6308: 6297: 6269: 6265: 6259: 6248: 6235: 6234: 6213: 6209: 6185: 6181: 6174: 6159: 6155: 6152: 6151: 6127: 6123: 6116: 6101: 6097: 6093: 6091: 6088: 6087: 6060:natural numbers 6033: 6006: 6002: 5994: 5990: 5976: 5972: 5968: 5931: 5928: 5927: 5887: 5879: 5876: 5865: 5852: 5851: 5849: 5847: 5844: 5843: 5839: 5831: 5827: 5789: 5785: 5783: 5780: 5779: 5760: 5748: 5740: 5734: 5733: 5731: 5720: 5708: 5702: 5701: 5699: 5684: 5680: 5678: 5675: 5674: 5673:, specifically 5645: 5642: 5641: 5618: 5606: 5600: 5599: 5598: 5589: 5582: 5569: 5568: 5567: 5566: 5560: 5549: 5543: 5540: 5539: 5535: 5534:. For example, 5528: 5524: 5494: 5476: 5470: 5469: 5468: 5461: 5443: 5437: 5436: 5435: 5434: 5428: 5410: 5404: 5403: 5401: 5389: 5385: 5370: 5359: 5341: 5338: 5337: 5316: 5313: 5312: 5292: 5277: 5270: 5266: 5265: 5253: 5241: 5238: 5237: 5221: 5218: 5217: 5184: 5164: 5162: 5158: 5157: 5146: 5141: 5124: 5120: 5105: 5101: 5100: 5098: 5096: 5093: 5092: 5071: 5064: 5060: 5047: 5045: 5030: 5026: 5022: 5017: 5011: 5000: 4984: 4980: 4966: 4962: 4958: 4953: 4941: 4930: 4914: 4910: 4892: 4881: 4877: 4865: 4861: 4859: 4857: 4854: 4853: 4814: 4806: 4804: 4798: 4787: 4773: 4760: 4759: 4758: 4752: 4741: 4728: 4724: 4722: 4719: 4718: 4696: 4693: 4692: 4675: 4671: 4669: 4666: 4665: 4649: 4629: 4625: 4623: 4620: 4619: 4603: 4600: 4599: 4594: 4583: +1 into 4575:The numbers of 4558: 4541:random variable 4518: 4515: 4514: 4450: 4447: 4446: 4419: 4416: 4415: 4384: 4376: 4373: 4372: 4352: 4349: 4348: 4331: 4327: 4325: 4322: 4321: 4318: 4288: 4275: 4274: 4272: 4270: 4267: 4266: 4249: 4246: 4245: 4229: 4226: 4225: 4182: 4174: 4172: 4162: 4149: 4148: 4147: 4137: 4133: 4127: 4124: 4123: 4114: 4109: 4104: 4080: 4078: 4075: 4074: 4055: 4052: 4051: 4035: 4032: 4031: 4015: 4012: 4011: 3995: 3992: 3991: 3984: 3963: 3960: 3959: 3942: 3938: 3936: 3933: 3932: 3916: 3913: 3912: 3896: 3893: 3892: 3876: 3873: 3872: 3850: 3846: 3837: 3833: 3811: 3798: 3797: 3796: 3786: 3774: 3765: 3764: 3763: 3747: 3734: 3733: 3732: 3722: 3709: 3708: 3707: 3697: 3684: 3683: 3682: 3676: 3665: 3659: 3656: 3655: 3627: 3624: 3623: 3603: 3599: 3581: 3572: 3568: 3559: 3555: 3541: 3538: 3537: 3506: 3502: 3488: 3485: 3484: 3467: 3463: 3461: 3458: 3457: 3434: 3430: 3428: 3425: 3424: 3407: 3403: 3388: 3384: 3382: 3379: 3378: 3355: 3351: 3337: 3334: 3333: 3316: 3312: 3310: 3307: 3306: 3272: 3269: 3268: 3252: 3251: 3236: 3232: 3223: 3219: 3210: 3206: 3191: 3187: 3178: 3167: 3154: 3150: 3141: 3140: 3128: 3124: 3118: 3114: 3105: 3101: 3092: 3088: 3073: 3069: 3060: 3049: 3036: 3032: 3026: 3022: 3013: 3012: 3006: 3002: 2996: 2992: 2986: 2975: 2962: 2958: 2952: 2948: 2933: 2929: 2923: 2919: 2910: 2906: 2894: 2890: 2884: 2873: 2860: 2859: 2853: 2849: 2843: 2839: 2833: 2822: 2809: 2805: 2793: 2789: 2777: 2766: 2755: 2749: 2745: 2739: 2735: 2729: 2718: 2699: 2695: 2689: 2685: 2679: 2668: 2660: 2658: 2655: 2654: 2626: 2623: 2622: 2619: 2618: 2617: 2614: 2601: 2600: 2592: 2579: 2578: 2576: 2567: 2554: 2553: 2551: 2542: 2529: 2528: 2526: 2517: 2504: 2503: 2501: 2492: 2479: 2478: 2476: 2467: 2454: 2453: 2451: 2448: 2447: 2439: 2426: 2425: 2423: 2414: 2401: 2400: 2398: 2389: 2376: 2375: 2373: 2364: 2351: 2350: 2348: 2339: 2326: 2325: 2323: 2320: 2319: 2311: 2298: 2297: 2295: 2286: 2273: 2272: 2270: 2261: 2248: 2247: 2245: 2236: 2223: 2222: 2220: 2217: 2216: 2208: 2195: 2194: 2192: 2183: 2170: 2169: 2167: 2158: 2145: 2144: 2142: 2139: 2138: 2130: 2117: 2116: 2114: 2105: 2092: 2091: 2089: 2086: 2085: 2077: 2064: 2063: 2061: 2057: 2055: 2052: 2051: 2028: 2024: 2018: 2014: 2008: 1997: 1978: 1974: 1968: 1964: 1958: 1947: 1934: 1930: 1909: 1905: 1881: 1877: 1835: 1831: 1817: 1814: 1813: 1793: 1789: 1783: 1779: 1773: 1762: 1749: 1745: 1731: 1728: 1727: 1705: 1702: 1701: 1684: 1680: 1666: 1663: 1662: 1639: 1635: 1621: 1618: 1617: 1588: 1584: 1582: 1579: 1578: 1561: 1557: 1555: 1552: 1551: 1528: 1515: 1514: 1513: 1504: 1500: 1498: 1495: 1494: 1478: 1475: 1474: 1457: 1453: 1451: 1448: 1447: 1427: 1423: 1417: 1413: 1398: 1394: 1379: 1375: 1360: 1356: 1344: 1340: 1334: 1330: 1312: 1308: 1302: 1298: 1289: 1285: 1279: 1275: 1266: 1262: 1250: 1246: 1240: 1236: 1230: 1219: 1206: 1202: 1188: 1185: 1184: 1168: 1165: 1164: 1142: 1139: 1138: 1102: 1089: 1088: 1086: 1084: 1081: 1080: 1055: 1042: 1041: 1039: 1037: 1034: 1033: 1008: 995: 994: 992: 990: 987: 986: 966: 962: 956: 952: 947: 938: 934: 928: 924: 919: 910: 906: 900: 896: 891: 882: 878: 857: 853: 844: 840: 826: 823: 822: 807: 629: 592: 589: 588: 572: 569: 568: 550: 535: 529: 528: 527: 517: 505: 494: 488: 487: 486: 476: 463: 462: 461: 459: 456: 455: 428: 415: 414: 412: 410: 407: 406: 402: 398: 379: 366: 365: 363: 361: 358: 357: 341: 338: 337: 321: 318: 317: 306: 278: 275: 274: 252: 249: 248: 224:is an infinite 214: 213: 212: 209: 196: 195: 161: 160: 130: 129: 103: 102: 80: 79: 61: 60: 46: 45: 35: 34: 27: 25: 22: 21: 17: 12: 11: 5: 13364: 13354: 13353: 13348: 13343: 13326: 13325: 13323: 13322: 13312: 13301: 13298: 13297: 13295: 13294: 13286: 13278: 13270: 13265:Étienne Pascal 13261: 13259: 13255: 13254: 13252: 13251: 13241: 13230: 13228: 13224: 13223: 13221: 13220: 13218:Pascal's wager 13215: 13210: 13205: 13200: 13194: 13192: 13190: 13189: 13186: 13182: 13179: 13178: 13171: 13170: 13163: 13156: 13148: 13142: 13141: 13129: 13121: 13102: 13082: 13081:External links 13079: 13076: 13075: 13046: 13027:(5): 461–463, 13011: 12992:(5): 425–428, 12976: 12968: 12938: 12904: 12877: 12860: 12841: 12822:(6): 392–394, 12806: 12788: 12784: 12761: 12757: 12734: 12730: 12707: 12703: 12680: 12676: 12648: 12640: 12610: 12602: 12572: 12565: 12545: 12526:(2): 283–300, 12509: 12493: 12484: 12482:(p. 420). 12449:(6): 538–544, 12433: 12388: 12360: 12319: 12282: 12258: 12217: 12202: 12195: 12175: 12152: 12139:(3): 213–223. 12119: 12098: 12075: 12068: 12048: 12036: 12029: 12001: 11976: 11971: 11968: 11963: 11943: 11936: 11916: 11903: 11876:(3): 147–157, 11851: 11850: 11848: 11845: 11843: 11842: 11837: 11832: 11827: 11822: 11816: 11811: 11806: 11801: 11795: 11790: 11785: 11780: 11775: 11773:Euler triangle 11770: 11765: 11760: 11755: 11750: 11743: 11741: 11738: 11737: 11736: 11723: 11719: 11715: 11712: 11707: 11703: 11699: 11696: 11693: 11682: 11679: 11673: 11670: 11667: 11664: 11661: 11658: 11655: 11652: 11649: 11646: 11643: 11638: 11633: 11629: 11602: 11580: 11575: 11571: 11550: 11528: 11523: 11519: 11498: 11495: 11492: 11466: 11461: 11457: 11436: 11433: 11430: 11408: 11404: 11383: 11363: 11360: 11357: 11337: 11317: 11297: 11273: 11267: 11264: 11261: 11257: 11252: 11225: 11222: 11219: 11199: 11196: 11193: 11190: 11187: 11176: 11175: 11162: 11158: 11154: 11151: 11148: 11143: 11139: 11135: 11132: 11129: 11126: 11123: 11120: 11117: 11114: 11111: 11108: 11105: 11102: 11099: 11096: 11093: 11090: 11087: 11084: 11081: 11078: 11075: 11072: 11069: 11066: 11063: 11058: 11053: 11049: 11025: 11002: 10997: 10993: 10969: 10964: 10960: 10956: 10951: 10946: 10940: 10937: 10932: 10929: 10925: 10918: 10914: 10893: 10890: 10887: 10865: 10861: 10857: 10852: 10847: 10843: 10820: 10816: 10812: 10807: 10802: 10798: 10777: 10757: 10746: 10745: 10730: 10710: 10707: 10702: 10699: 10696: 10691: 10688: 10685: 10682: 10679: 10676: 10673: 10670: 10667: 10664: 10661: 10658: 10655: 10652: 10628: 10625: 10622: 10602: 10599: 10596: 10593: 10590: 10568: 10563: 10559: 10555: 10550: 10546: 10542: 10539: 10536: 10533: 10513: 10489: 10469: 10466: 10463: 10443: 10421: 10417: 10413: 10410: 10407: 10404: 10393: 10379: 10375: 10371: 10368: 10365: 10360: 10356: 10352: 10349: 10346: 10341: 10337: 10333: 10330: 10327: 10322: 10318: 10314: 10311: 10308: 10303: 10299: 10295: 10292: 10289: 10284: 10280: 10259: 10239: 10236: 10233: 10209: 10185: 10163: 10159: 10134: 10108: 10103: 10099: 10095: 10090: 10086: 10082: 10079: 10076: 10073: 10053: 10026: 10021: 10017: 10011: 10008: 10005: 10001: 9979: 9959: 9940: 9937: 9931:to the entire 9916: 9913: 9910: 9907: 9904: 9901: 9880: 9859: 9856: 9853: 9850: 9847: 9844: 9841: 9838: 9835: 9823: 9820: 9795: 9792: 9787: 9784: 9772: 9769: 9766: 9763: 9760: 9757: 9754: 9751: 9748: 9745: 9742: 9739: 9736: 9733: 9730: 9727: 9701: 9700: 9688: 9683: 9678: 9672: 9668: 9664: 9661: 9658: 9655: 9649: 9639: 9633: 9630: 9610: 9609: 9597: 9592: 9587: 9581: 9577: 9573: 9570: 9567: 9564: 9558: 9548: 9542: 9539: 9515:imaginary part 9478: 9468: 9421: 9418: 9405: 9402: 9397: 9393: 9343:-cube from an 9321: 9320: 9305: 9301: 9297: 9293: 9289: 9268: 9267: 9258: 9254: 9250: 9229: 9228: 9190: 9189: 9157: 9156: 9130: 9129: 9109: 9108: 9094: 9093: 9085: 9084: 9070: 9069: 9058: 9052: 9047: 9043: 9040: 9037: 9031: 9025: 9019: 9013: 9010: 9007: 9002: 8999: 8996: 8990: 8984: 8981: 8978: 8972: 8967: 8964: 8959: 8921: 8918: 8803: 8800: 8786: 8785: 8736: 8734: 8727: 8721: 8718: 8704: 8662: 8659: 8641: 8620: 8617: 8615: 8607: 8603: 8602: 8597: 8591: 8588: 8586: 8583: 8581: 8578: 8576: 8573: 8571: 8568: 8567: 8564: 8561: 8559: 8556: 8554: 8551: 8549: 8546: 8544: 8541: 8540: 8537: 8534: 8532: 8529: 8527: 8524: 8522: 8519: 8517: 8514: 8513: 8510: 8507: 8505: 8502: 8500: 8497: 8495: 8492: 8490: 8487: 8486: 8483: 8480: 8478: 8475: 8473: 8470: 8468: 8465: 8463: 8460: 8459: 8457: 8452: 8449: 8447: 8443: 8437: 8434: 8432: 8429: 8427: 8424: 8422: 8419: 8417: 8414: 8413: 8410: 8407: 8405: 8402: 8400: 8397: 8395: 8392: 8390: 8387: 8386: 8383: 8380: 8378: 8375: 8373: 8370: 8368: 8365: 8363: 8360: 8359: 8356: 8353: 8351: 8348: 8346: 8343: 8341: 8338: 8336: 8333: 8332: 8329: 8326: 8324: 8321: 8319: 8316: 8314: 8311: 8309: 8306: 8305: 8303: 8298: 8295: 8292: 8291: 8282: 8281: 8280: 8278: 8275: 8274: 8273: 8272: 8271: 8268: 8267: 8264: 8261: 8258: 8255: 8252: 8249: 8246: 8242: 8241: 8238: 8235: 8232: 8229: 8226: 8223: 8219: 8218: 8215: 8212: 8209: 8206: 8203: 8199: 8198: 8195: 8192: 8189: 8186: 8182: 8181: 8178: 8175: 8172: 8168: 8167: 8164: 8161: 8157: 8156: 8153: 8149: 8148: 8138: 8137: 8122: 8121: 8107: 8104: 8103: 8098: 8093: 8086: 8078: 8077: 8072: 8065: 8058: 8050: 8049: 8042: 8035: 8028: 8020: 8019: 8012: 8005: 7998: 7988: 7987: 7986: 7985: 7973: 7972: 7947: 7944: 7916: 7911: 7908: 7903: 7896: 7889: 7884: 7881: 7876: 7869: 7862: 7857: 7854: 7849: 7825: 7822: 7816: 7813: 7807: 7804: 7801: 7794: 7789: 7786: 7781: 7774: 7771: 7768: 7762: 7759: 7753: 7750: 7747: 7740: 7735: 7732: 7727: 7720: 7717: 7714: 7707: 7702: 7699: 7694: 7670: 7667: 7661: 7658: 7652: 7646: 7643: 7637: 7631: 7628: 7601: 7596: 7593: 7588: 7573: 7572: 7561: 7556: 7552: 7549: 7546: 7540: 7534: 7528: 7525: 7522: 7517: 7514: 7511: 7508: 7505: 7499: 7493: 7487: 7482: 7478: 7475: 7472: 7466: 7440: 7437: 7430: 7425: 7422: 7417: 7393: 7390: 7387: 7380: 7375: 7371: 7368: 7365: 7359: 7352: 7345: 7340: 7336: 7333: 7330: 7324: 7317: 7310: 7305: 7302: 7297: 7270: 7267: 7261: 7258: 7252: 7249: 7246: 7239: 7234: 7231: 7226: 7202: 7199: 7193: 7190: 7184: 7181: 7178: 7171: 7166: 7163: 7158: 7134: 7131: 7124: 7119: 7116: 7111: 7084: 7081: 7055: 7052: 7026: 7023: 6997: 6994: 6968: 6965: 6951: 6950: 6939: 6934: 6930: 6927: 6924: 6921: 6918: 6912: 6906: 6900: 6897: 6894: 6890: 6885: 6879: 6873: 6868: 6865: 6860: 6834: 6831: 6824: 6819: 6816: 6811: 6783: 6778: 6775: 6770: 6763: 6760: 6757: 6750: 6745: 6742: 6737: 6730: 6723: 6718: 6715: 6710: 6686: 6671: 6668: 6660: − 1 6656: 6641: 6630: 6617: 6603: 6597: 6577:. Construct a 6568: 6559: 6533: 6527: 6522: 6518: 6515: 6512: 6509: 6506: 6500: 6494: 6488: 6485: 6479: 6476: 6473: 6469: 6463: 6460: 6457: 6454: 6451: 6448: 6443: 6440: 6437: 6432: 6429: 6426: 6422: 6415: 6412: 6408: 6403: 6400: 6397: 6394: 6389: 6385: 6359: 6358: 6357: 6356: 6341: 6338: 6335: 6332: 6329: 6326: 6321: 6317: 6311: 6306: 6303: 6300: 6296: 6292: 6289: 6286: 6283: 6278: 6275: 6272: 6268: 6262: 6257: 6254: 6251: 6247: 6243: 6240: 6238: 6236: 6233: 6230: 6227: 6222: 6219: 6216: 6212: 6208: 6205: 6202: 6199: 6196: 6193: 6188: 6184: 6180: 6177: 6175: 6173: 6170: 6167: 6162: 6158: 6154: 6153: 6150: 6147: 6144: 6141: 6138: 6135: 6130: 6126: 6122: 6119: 6117: 6115: 6112: 6109: 6104: 6100: 6096: 6095: 6082: 6081: 6070: 6063: 6056: 6051:of simplices: 6039:Derivation of 6032: 6029: 6028: 6027: 6021: 6014: 5980: 5956: 5953: 5950: 5947: 5944: 5941: 5938: 5935: 5911: 5908: 5905: 5902: 5899: 5896: 5893: 5890: 5885: 5882: 5875: 5868: 5863: 5860: 5855: 5826:In a row 5824: 5812: 5809: 5806: 5803: 5800: 5797: 5792: 5788: 5763: 5757: 5754: 5751: 5746: 5743: 5737: 5730: 5723: 5718: 5714: 5711: 5705: 5698: 5693: 5690: 5687: 5683: 5671:Catalan number 5658: 5655: 5652: 5649: 5638: 5627: 5621: 5616: 5612: 5609: 5603: 5597: 5592: 5585: 5580: 5577: 5572: 5563: 5558: 5555: 5552: 5548: 5521: 5514: 5497: 5492: 5488: 5485: 5482: 5479: 5473: 5464: 5459: 5455: 5452: 5449: 5446: 5440: 5431: 5426: 5422: 5419: 5416: 5413: 5407: 5398: 5395: 5392: 5388: 5384: 5381: 5378: 5373: 5368: 5365: 5362: 5358: 5354: 5351: 5348: 5345: 5320: 5309: 5295: 5290: 5284: 5281: 5276: 5273: 5269: 5262: 5259: 5256: 5252: 5248: 5245: 5225: 5204: 5201: 5198: 5192: 5187: 5182: 5177: 5173: 5170: 5167: 5161: 5156: 5149: 5144: 5140: 5133: 5130: 5127: 5123: 5119: 5114: 5111: 5108: 5104: 5077: 5074: 5067: 5063: 5059: 5056: 5053: 5050: 5044: 5033: 5029: 5025: 5021: 5014: 5009: 5006: 5003: 4999: 4993: 4990: 4987: 4983: 4979: 4969: 4965: 4961: 4957: 4950: 4947: 4944: 4939: 4936: 4933: 4929: 4923: 4920: 4917: 4913: 4909: 4906: 4903: 4900: 4897: 4891: 4884: 4880: 4874: 4871: 4868: 4864: 4838: 4835: 4832: 4829: 4826: 4823: 4820: 4817: 4812: 4809: 4801: 4796: 4793: 4790: 4786: 4782: 4776: 4771: 4768: 4763: 4755: 4750: 4747: 4744: 4740: 4736: 4731: 4727: 4706: 4703: 4700: 4678: 4674: 4646: 4632: 4628: 4607: 4593: 4590: 4557: 4554: 4528: 4525: 4522: 4502: 4499: 4496: 4493: 4490: 4487: 4484: 4481: 4478: 4475: 4472: 4469: 4466: 4463: 4460: 4457: 4454: 4423: 4392: 4389: 4383: 4380: 4356: 4334: 4330: 4317: 4314: 4301: 4298: 4291: 4286: 4283: 4278: 4253: 4233: 4222: 4221: 4206: 4203: 4200: 4197: 4194: 4191: 4188: 4185: 4180: 4177: 4171: 4165: 4160: 4157: 4152: 4146: 4140: 4136: 4130: 4126: 4122: 4117: 4112: 4107: 4102: 4099: 4096: 4093: 4090: 4087: 4083: 4059: 4039: 4019: 3999: 3983: 3980: 3967: 3945: 3941: 3920: 3900: 3880: 3858: 3853: 3849: 3845: 3840: 3836: 3832: 3829: 3826: 3823: 3820: 3814: 3809: 3806: 3801: 3795: 3789: 3783: 3780: 3777: 3773: 3768: 3762: 3759: 3756: 3750: 3745: 3742: 3737: 3731: 3725: 3720: 3717: 3712: 3706: 3700: 3695: 3692: 3687: 3679: 3674: 3671: 3668: 3664: 3643: 3640: 3637: 3634: 3631: 3606: 3602: 3598: 3595: 3589: 3586: 3580: 3575: 3571: 3567: 3562: 3558: 3554: 3551: 3548: 3545: 3509: 3505: 3501: 3498: 3495: 3492: 3470: 3466: 3443: 3440: 3437: 3433: 3410: 3406: 3402: 3397: 3394: 3391: 3387: 3364: 3361: 3358: 3354: 3350: 3347: 3344: 3341: 3319: 3315: 3294: 3291: 3288: 3285: 3282: 3279: 3276: 3250: 3245: 3242: 3239: 3235: 3231: 3226: 3222: 3218: 3213: 3209: 3205: 3200: 3197: 3194: 3190: 3186: 3181: 3176: 3173: 3170: 3166: 3162: 3157: 3153: 3149: 3146: 3144: 3142: 3137: 3134: 3131: 3127: 3121: 3117: 3113: 3108: 3104: 3100: 3095: 3091: 3087: 3082: 3079: 3076: 3072: 3068: 3063: 3058: 3055: 3052: 3048: 3044: 3039: 3035: 3029: 3025: 3021: 3018: 3016: 3014: 3009: 3005: 2999: 2995: 2989: 2984: 2981: 2978: 2974: 2970: 2965: 2961: 2955: 2951: 2947: 2942: 2939: 2936: 2932: 2926: 2922: 2918: 2913: 2909: 2903: 2900: 2897: 2893: 2887: 2882: 2879: 2876: 2872: 2868: 2865: 2863: 2861: 2856: 2852: 2846: 2842: 2836: 2831: 2828: 2825: 2821: 2817: 2812: 2808: 2802: 2799: 2796: 2792: 2786: 2783: 2780: 2775: 2772: 2769: 2765: 2761: 2758: 2756: 2752: 2748: 2742: 2738: 2732: 2727: 2724: 2721: 2717: 2713: 2708: 2705: 2702: 2698: 2692: 2688: 2682: 2677: 2674: 2671: 2667: 2663: 2662: 2642: 2639: 2636: 2633: 2630: 2615: 2595: 2590: 2587: 2582: 2570: 2565: 2562: 2557: 2545: 2540: 2537: 2532: 2520: 2515: 2512: 2507: 2495: 2490: 2487: 2482: 2470: 2465: 2462: 2457: 2450: 2449: 2442: 2437: 2434: 2429: 2417: 2412: 2409: 2404: 2392: 2387: 2384: 2379: 2367: 2362: 2359: 2354: 2342: 2337: 2334: 2329: 2322: 2321: 2314: 2309: 2306: 2301: 2289: 2284: 2281: 2276: 2264: 2259: 2256: 2251: 2239: 2234: 2231: 2226: 2219: 2218: 2211: 2206: 2203: 2198: 2186: 2181: 2178: 2173: 2161: 2156: 2153: 2148: 2141: 2140: 2133: 2128: 2125: 2120: 2108: 2103: 2100: 2095: 2088: 2087: 2080: 2075: 2072: 2067: 2060: 2059: 2050: 2049: 2048: 2036: 2031: 2027: 2021: 2017: 2011: 2006: 2003: 2000: 1996: 1992: 1987: 1984: 1981: 1977: 1971: 1967: 1961: 1956: 1953: 1950: 1946: 1942: 1937: 1933: 1929: 1926: 1923: 1920: 1917: 1912: 1908: 1904: 1901: 1898: 1895: 1892: 1889: 1884: 1880: 1876: 1873: 1870: 1867: 1864: 1861: 1858: 1855: 1852: 1849: 1844: 1841: 1838: 1834: 1830: 1827: 1824: 1821: 1801: 1796: 1792: 1786: 1782: 1776: 1771: 1768: 1765: 1761: 1757: 1752: 1748: 1744: 1741: 1738: 1735: 1715: 1712: 1709: 1687: 1683: 1679: 1676: 1673: 1670: 1648: 1645: 1642: 1638: 1634: 1631: 1628: 1625: 1602: 1597: 1594: 1591: 1587: 1564: 1560: 1537: 1531: 1526: 1523: 1518: 1512: 1507: 1503: 1482: 1460: 1456: 1435: 1430: 1426: 1420: 1416: 1412: 1407: 1404: 1401: 1397: 1393: 1388: 1385: 1382: 1378: 1374: 1371: 1368: 1363: 1359: 1353: 1350: 1347: 1343: 1337: 1333: 1329: 1326: 1321: 1318: 1315: 1311: 1305: 1301: 1297: 1292: 1288: 1282: 1278: 1274: 1269: 1265: 1259: 1256: 1253: 1249: 1243: 1239: 1233: 1228: 1225: 1222: 1218: 1214: 1209: 1205: 1201: 1198: 1195: 1192: 1172: 1152: 1149: 1146: 1115: 1112: 1105: 1100: 1097: 1092: 1068: 1065: 1058: 1053: 1050: 1045: 1021: 1018: 1011: 1006: 1003: 998: 974: 969: 965: 959: 955: 950: 946: 941: 937: 931: 927: 922: 918: 913: 909: 903: 899: 894: 890: 885: 881: 877: 874: 871: 868: 865: 860: 856: 852: 847: 843: 839: 836: 833: 830: 806: 803: 746:Michael Stifel 738:Petrus Apianus 628: 625: 608: 605: 602: 599: 596: 576: 553: 548: 544: 541: 538: 532: 526: 520: 514: 511: 508: 503: 500: 497: 491: 485: 479: 474: 471: 466: 441: 438: 431: 426: 423: 418: 397:, pronounced " 382: 377: 374: 369: 345: 325: 305: 302: 288: 285: 282: 262: 259: 256: 242:mathematicians 210: 194: 190: 186: 182: 178: 174: 170: 166: 163: 162: 159: 155: 151: 147: 143: 139: 135: 132: 131: 128: 124: 120: 116: 112: 108: 105: 104: 101: 97: 93: 89: 85: 82: 81: 78: 74: 70: 66: 63: 62: 59: 55: 51: 48: 47: 44: 40: 37: 36: 33: 30: 29: 20: 19: 18: 15: 9: 6: 4: 3: 2: 13363: 13352: 13349: 13347: 13346:Blaise Pascal 13344: 13342: 13339: 13338: 13336: 13321: 13313: 13311: 13303: 13302: 13299: 13290: 13287: 13282: 13279: 13274: 13271: 13266: 13263: 13262: 13260: 13256: 13247: 13246: 13242: 13237: 13236: 13232: 13231: 13229: 13225: 13219: 13216: 13214: 13211: 13209: 13206: 13204: 13201: 13199: 13196: 13195: 13193: 13187: 13184: 13183: 13180: 13176: 13175:Blaise Pascal 13169: 13164: 13162: 13157: 13155: 13150: 13149: 13146: 13140: 13138: 13133: 13130: 13128: 13125: 13122: 13117: 13116: 13111: 13108: 13103: 13099: 13095: 13094: 13089: 13085: 13084: 13070: 13065: 13061: 13057: 13050: 13042: 13038: 13034: 13030: 13026: 13022: 13015: 13007: 13003: 12999: 12995: 12991: 12987: 12980: 12971: 12969:9780080372372 12965: 12961: 12957: 12953: 12949: 12942: 12934: 12930: 12926: 12922: 12915: 12908: 12900: 12896: 12892: 12888: 12881: 12873: 12872: 12864: 12856: 12852: 12845: 12837: 12833: 12829: 12825: 12821: 12817: 12810: 12802: 12786: 12782: 12759: 12755: 12732: 12728: 12705: 12701: 12678: 12674: 12663: 12659: 12652: 12643: 12641:9780080372372 12637: 12633: 12629: 12625: 12621: 12614: 12605: 12603:9780080372372 12599: 12595: 12591: 12587: 12583: 12576: 12568: 12566:9780323139595 12562: 12558: 12557: 12549: 12541: 12537: 12533: 12529: 12525: 12521: 12513: 12506: 12505: 12497: 12488: 12481: 12477: 12476:Édouard Lucas 12472: 12468: 12464: 12460: 12456: 12452: 12448: 12444: 12437: 12429: 12425: 12421: 12417: 12413: 12409: 12405: 12401: 12400: 12392: 12385: 12381: 12377: 12373: 12372: 12364: 12356: 12352: 12348: 12344: 12340: 12336: 12335: 12330: 12323: 12315: 12311: 12307: 12303: 12299: 12295: 12294: 12286: 12272: 12268: 12262: 12253: 12249: 12245: 12241: 12237: 12233: 12232: 12227: 12226:Fowler, David 12221: 12215: 12211: 12206: 12198: 12196:9780538737586 12192: 12188: 12187: 12179: 12170: 12163: 12161: 12159: 12157: 12147: 12142: 12138: 12134: 12130: 12123: 12116: 12112: 12108: 12102: 12094: 12090: 12086: 12079: 12071: 12069:9783642367366 12065: 12061: 12060: 12052: 12045: 12040: 12032: 12030:9781402045592 12026: 12022: 12018: 12014: 12013: 12005: 11998: 11994: 11969: 11966: 11947: 11939: 11933: 11929: 11928: 11920: 11913: 11907: 11899: 11895: 11891: 11887: 11883: 11879: 11875: 11871: 11870: 11865: 11859: 11857: 11852: 11841: 11838: 11836: 11833: 11831: 11828: 11826: 11823: 11820: 11817: 11815: 11812: 11810: 11807: 11805: 11804:Pascal matrix 11802: 11799: 11796: 11794: 11791: 11789: 11786: 11784: 11781: 11779: 11776: 11774: 11771: 11769: 11766: 11764: 11761: 11759: 11756: 11754: 11753:Bell triangle 11751: 11748: 11745: 11744: 11721: 11717: 11713: 11710: 11705: 11701: 11697: 11694: 11691: 11680: 11677: 11671: 11668: 11665: 11662: 11659: 11656: 11653: 11650: 11647: 11644: 11641: 11636: 11631: 11627: 11619: 11618: 11617: 11615: 11600: 11578: 11573: 11569: 11548: 11526: 11521: 11517: 11496: 11493: 11490: 11482: 11464: 11459: 11455: 11434: 11431: 11428: 11406: 11402: 11381: 11361: 11358: 11355: 11335: 11315: 11295: 11265: 11262: 11259: 11255: 11239: 11223: 11220: 11217: 11197: 11194: 11191: 11188: 11185: 11160: 11156: 11152: 11149: 11146: 11141: 11137: 11133: 11130: 11127: 11124: 11121: 11118: 11115: 11112: 11109: 11106: 11103: 11100: 11097: 11094: 11091: 11088: 11085: 11082: 11079: 11076: 11073: 11070: 11067: 11064: 11061: 11056: 11051: 11047: 11039: 11038: 11037: 11023: 11015: 11000: 10995: 10991: 10967: 10962: 10958: 10954: 10949: 10944: 10938: 10935: 10930: 10927: 10923: 10916: 10912: 10891: 10888: 10885: 10863: 10859: 10855: 10850: 10845: 10841: 10818: 10814: 10810: 10805: 10800: 10796: 10775: 10755: 10743: 10728: 10708: 10705: 10683: 10680: 10677: 10671: 10668: 10665: 10662: 10659: 10653: 10650: 10642: 10626: 10623: 10620: 10600: 10597: 10594: 10591: 10588: 10566: 10561: 10557: 10553: 10548: 10540: 10537: 10534: 10511: 10503: 10487: 10467: 10464: 10461: 10441: 10419: 10411: 10408: 10405: 10394: 10377: 10373: 10369: 10366: 10363: 10358: 10354: 10350: 10347: 10344: 10339: 10335: 10331: 10328: 10325: 10320: 10316: 10312: 10309: 10306: 10301: 10297: 10293: 10290: 10287: 10282: 10278: 10257: 10237: 10234: 10231: 10223: 10207: 10199: 10183: 10161: 10157: 10148: 10132: 10124: 10123: 10122: 10106: 10101: 10097: 10093: 10088: 10080: 10077: 10074: 10051: 10043: 10039: 10024: 10019: 10015: 10003: 9977: 9957: 9950: 9945: 9936: 9934: 9933:complex plane 9930: 9911: 9908: 9905: 9854: 9851: 9848: 9839: 9836: 9833: 9819: 9817: 9816: 9811: 9807: 9806: 9801: 9791: 9783: 9770: 9767: 9764: 9761: 9758: 9755: 9752: 9749: 9746: 9743: 9740: 9737: 9734: 9731: 9728: 9725: 9717: 9713: 9708: 9706: 9686: 9681: 9676: 9670: 9662: 9656: 9653: 9647: 9637: 9619: 9618: 9617: 9615: 9595: 9590: 9585: 9579: 9571: 9565: 9562: 9556: 9546: 9528: 9527: 9526: 9524: 9520: 9519:step function 9516: 9512: 9508: 9504: 9500: 9496: 9492: 9488: 9484: 9477: 9473: 9467: 9465: 9461: 9443: 9439: 9435: 9431: 9427: 9417: 9403: 9400: 9395: 9391: 9378: 9373: 9371: 9367: 9363: 9358: 9354: 9348: 9342: 9337: 9335: 9331: 9303: 9299: 9295: 9291: 9287: 9256: 9252: 9248: 9075: 9074: 9073: 9056: 9045: 9041: 9038: 9035: 9023: 9011: 9008: 9005: 9000: 8997: 8994: 8982: 8979: 8976: 8965: 8962: 8947: 8946: 8945: 8941: 8936:, instead of 8933: 8927: 8917: 8914: 8899: 8884: 8870: 8857: 8853: 8848: 8846: 8842: 8838: 8832: 8826: 8821: 8819: 8814: 8810: 8799: 8797: 8793: 8782: 8779: 8771: 8761: 8757: 8753: 8747: 8746: 8742: 8737:This section 8735: 8731: 8726: 8725: 8717: 8692: 8688: 8672: 8668: 8658: 8656: 8650: 8649:Pascal matrix 8618: 8616: 8605: 8595: 8589: 8584: 8579: 8574: 8569: 8562: 8557: 8552: 8547: 8542: 8535: 8530: 8525: 8520: 8515: 8508: 8503: 8498: 8493: 8488: 8481: 8476: 8471: 8466: 8461: 8455: 8450: 8448: 8441: 8435: 8430: 8425: 8420: 8415: 8408: 8403: 8398: 8393: 8388: 8381: 8376: 8371: 8366: 8361: 8354: 8349: 8344: 8339: 8334: 8327: 8322: 8317: 8312: 8307: 8301: 8296: 8293: 8265: 8262: 8259: 8256: 8253: 8250: 8247: 8244: 8236: 8227: 8224: 8210: 8207: 8201: 8196: 8193: 8187: 8176: 8159: 8154: 8151: 8142: 8141: 8140: 8139: 8135: 8131: 8130: 8126: 8119: 8115: 8114: 8110: 8099: 8094: 8091: 8087: 8084: 8080: 8079: 8073: 8070: 8066: 8063: 8059: 8056: 8052: 8051: 8047: 8043: 8040: 8036: 8033: 8029: 8026: 8022: 8021: 8017: 8013: 8010: 8006: 8003: 7999: 7996: 7992: 7991: 7983: 7979: 7975: 7974: 7970: 7966: 7963:known as the 7962: 7958: 7957: 7952: 7940: 7936: 7932: 7909: 7906: 7894: 7882: 7879: 7867: 7855: 7852: 7823: 7820: 7814: 7811: 7805: 7802: 7799: 7787: 7784: 7772: 7769: 7766: 7760: 7757: 7751: 7748: 7745: 7733: 7730: 7718: 7715: 7712: 7700: 7697: 7668: 7665: 7659: 7656: 7650: 7644: 7641: 7635: 7629: 7626: 7594: 7591: 7559: 7554: 7550: 7547: 7544: 7538: 7526: 7523: 7520: 7515: 7512: 7509: 7506: 7503: 7491: 7480: 7476: 7473: 7470: 7454: 7453: 7452: 7438: 7435: 7423: 7420: 7391: 7388: 7385: 7373: 7369: 7366: 7363: 7350: 7338: 7334: 7331: 7328: 7315: 7303: 7300: 7282: 7268: 7265: 7259: 7256: 7250: 7247: 7244: 7232: 7229: 7200: 7197: 7191: 7188: 7182: 7179: 7176: 7164: 7161: 7132: 7129: 7117: 7114: 7082: 7079: 7053: 7050: 7024: 7021: 6995: 6992: 6966: 6963: 6937: 6932: 6928: 6925: 6922: 6919: 6916: 6910: 6898: 6895: 6892: 6888: 6877: 6866: 6863: 6848: 6847: 6846: 6832: 6829: 6817: 6814: 6798:, begin with 6776: 6773: 6761: 6758: 6755: 6743: 6740: 6728: 6716: 6713: 6684: 6675: 6667: 6665: 6659: 6655: 6651: 6647: 6640: 6636: 6629: 6625: 6620: 6615: 6611: 6606: 6600: 6596: 6592: 6588: 6584: 6580: 6576: 6571: 6567: 6562: 6558: 6553: 6551: 6547: 6531: 6520: 6516: 6513: 6510: 6507: 6504: 6492: 6486: 6483: 6474: 6467: 6461: 6455: 6452: 6449: 6441: 6438: 6435: 6430: 6427: 6424: 6420: 6413: 6410: 6406: 6401: 6395: 6387: 6383: 6373: 6371: 6368: 6364: 6339: 6333: 6330: 6327: 6319: 6315: 6309: 6304: 6301: 6298: 6294: 6290: 6284: 6276: 6273: 6270: 6266: 6260: 6255: 6252: 6249: 6245: 6241: 6239: 6228: 6220: 6217: 6214: 6210: 6206: 6200: 6197: 6194: 6186: 6182: 6178: 6176: 6168: 6160: 6156: 6148: 6145: 6142: 6136: 6128: 6124: 6120: 6118: 6110: 6102: 6098: 6086: 6085: 6084: 6083: 6079: 6075: 6071: 6068: 6064: 6061: 6057: 6054: 6053: 6052: 6050: 6042: 6037: 6025: 6022: 6019: 6015: 6012: 6000: 5988: 5984: 5981: 5954: 5948: 5945: 5942: 5936: 5933: 5909: 5903: 5900: 5897: 5891: 5888: 5883: 5880: 5873: 5861: 5858: 5837: 5825: 5810: 5807: 5804: 5801: 5798: 5795: 5790: 5786: 5755: 5752: 5749: 5744: 5741: 5728: 5716: 5712: 5709: 5696: 5691: 5688: 5685: 5681: 5672: 5656: 5653: 5650: 5647: 5639: 5625: 5614: 5610: 5607: 5595: 5590: 5578: 5575: 5561: 5556: 5553: 5550: 5546: 5532: 5522: 5519: 5515: 5490: 5486: 5483: 5480: 5477: 5457: 5453: 5450: 5447: 5444: 5424: 5420: 5417: 5414: 5411: 5396: 5393: 5390: 5382: 5379: 5366: 5363: 5360: 5356: 5352: 5349: 5346: 5343: 5335: 5331: 5318: 5310: 5293: 5288: 5282: 5279: 5274: 5271: 5267: 5254: 5246: 5243: 5236: 5223: 5202: 5199: 5196: 5190: 5185: 5180: 5175: 5171: 5168: 5165: 5159: 5154: 5147: 5142: 5138: 5131: 5128: 5125: 5121: 5117: 5112: 5109: 5106: 5102: 5075: 5072: 5065: 5057: 5054: 5051: 5042: 5031: 5027: 5023: 5019: 5012: 5007: 5004: 5001: 4997: 4991: 4988: 4985: 4981: 4977: 4967: 4963: 4959: 4955: 4948: 4945: 4942: 4937: 4934: 4931: 4927: 4921: 4918: 4915: 4911: 4904: 4901: 4898: 4889: 4882: 4878: 4872: 4869: 4866: 4862: 4836: 4830: 4827: 4824: 4818: 4815: 4810: 4807: 4799: 4794: 4791: 4788: 4784: 4780: 4769: 4766: 4753: 4748: 4745: 4742: 4738: 4734: 4729: 4725: 4704: 4701: 4698: 4676: 4672: 4663: 4662: 4657: 4652: 4647: 4630: 4626: 4605: 4596: 4595: 4586: 4582: 4578: 4573: 4565: 4561: 4553: 4551: 4546: 4542: 4526: 4523: 4520: 4497: 4494: 4491: 4488: 4485: 4482: 4479: 4476: 4473: 4470: 4467: 4464: 4461: 4458: 4455: 4444: 4439: 4437: 4421: 4413: 4409: 4390: 4387: 4381: 4378: 4370: 4354: 4332: 4328: 4313: 4299: 4296: 4284: 4281: 4251: 4231: 4204: 4198: 4195: 4192: 4186: 4183: 4178: 4175: 4169: 4158: 4155: 4144: 4138: 4134: 4128: 4125: 4120: 4115: 4110: 4100: 4094: 4091: 4088: 4073: 4072: 4071: 4057: 4037: 4017: 3997: 3989: 3979: 3965: 3943: 3939: 3918: 3898: 3878: 3869: 3856: 3851: 3847: 3843: 3838: 3830: 3827: 3824: 3818: 3807: 3804: 3793: 3781: 3778: 3775: 3771: 3760: 3757: 3754: 3743: 3740: 3729: 3718: 3715: 3704: 3693: 3690: 3677: 3672: 3669: 3666: 3662: 3641: 3638: 3635: 3632: 3629: 3620: 3604: 3596: 3593: 3587: 3584: 3573: 3569: 3565: 3560: 3552: 3549: 3546: 3534: 3532: 3528: 3523: 3507: 3499: 3496: 3493: 3468: 3464: 3441: 3438: 3435: 3431: 3408: 3404: 3400: 3395: 3392: 3389: 3385: 3362: 3359: 3356: 3348: 3345: 3342: 3317: 3313: 3292: 3289: 3286: 3283: 3280: 3277: 3274: 3265: 3248: 3243: 3240: 3237: 3233: 3229: 3224: 3220: 3211: 3207: 3203: 3198: 3195: 3192: 3188: 3179: 3174: 3171: 3168: 3164: 3160: 3155: 3151: 3147: 3145: 3135: 3132: 3129: 3125: 3119: 3115: 3111: 3106: 3102: 3093: 3089: 3085: 3080: 3077: 3074: 3070: 3061: 3056: 3053: 3050: 3046: 3042: 3037: 3033: 3027: 3023: 3019: 3017: 3007: 3003: 2997: 2993: 2987: 2982: 2979: 2976: 2972: 2968: 2963: 2959: 2953: 2949: 2945: 2940: 2937: 2934: 2930: 2924: 2920: 2916: 2911: 2907: 2901: 2898: 2895: 2891: 2885: 2880: 2877: 2874: 2870: 2866: 2864: 2854: 2850: 2844: 2840: 2834: 2829: 2826: 2823: 2819: 2815: 2810: 2806: 2800: 2797: 2794: 2790: 2784: 2781: 2778: 2773: 2770: 2767: 2763: 2759: 2757: 2750: 2746: 2740: 2736: 2730: 2725: 2722: 2719: 2715: 2711: 2706: 2703: 2700: 2696: 2690: 2686: 2680: 2675: 2672: 2669: 2665: 2640: 2637: 2634: 2631: 2628: 2588: 2585: 2563: 2560: 2538: 2535: 2513: 2510: 2488: 2485: 2463: 2460: 2435: 2432: 2410: 2407: 2385: 2382: 2360: 2357: 2335: 2332: 2307: 2304: 2282: 2279: 2257: 2254: 2232: 2229: 2204: 2201: 2179: 2176: 2154: 2151: 2126: 2123: 2101: 2098: 2073: 2070: 2047: 2034: 2029: 2025: 2019: 2015: 2009: 2004: 2001: 1998: 1994: 1990: 1985: 1982: 1979: 1975: 1969: 1965: 1959: 1954: 1951: 1948: 1944: 1940: 1935: 1927: 1924: 1921: 1915: 1910: 1902: 1899: 1896: 1890: 1887: 1882: 1874: 1871: 1868: 1859: 1856: 1853: 1847: 1842: 1839: 1836: 1828: 1825: 1822: 1799: 1794: 1790: 1784: 1780: 1774: 1769: 1766: 1763: 1759: 1755: 1750: 1742: 1739: 1736: 1713: 1710: 1707: 1685: 1677: 1674: 1671: 1646: 1643: 1640: 1632: 1629: 1626: 1614: 1613:, and so on. 1600: 1595: 1592: 1589: 1585: 1562: 1558: 1548: 1535: 1524: 1521: 1510: 1505: 1501: 1480: 1458: 1454: 1433: 1428: 1424: 1418: 1414: 1410: 1405: 1402: 1399: 1395: 1391: 1386: 1383: 1380: 1376: 1372: 1369: 1366: 1361: 1357: 1351: 1348: 1345: 1341: 1335: 1331: 1327: 1324: 1319: 1316: 1313: 1309: 1303: 1299: 1295: 1290: 1286: 1280: 1276: 1272: 1267: 1263: 1257: 1254: 1251: 1247: 1241: 1237: 1231: 1226: 1223: 1220: 1216: 1212: 1207: 1199: 1196: 1193: 1170: 1150: 1147: 1144: 1136: 1132: 1127: 1113: 1110: 1098: 1095: 1066: 1063: 1051: 1048: 1019: 1016: 1004: 1001: 972: 967: 963: 957: 953: 944: 939: 935: 929: 925: 916: 911: 907: 901: 897: 888: 883: 879: 875: 872: 869: 866: 863: 858: 854: 850: 845: 837: 834: 831: 820: 811: 802: 799: 794: 789: 784: 780: 776: 771: 765: 763: 759: 755: 751: 747: 743: 739: 735: 731: 727: 722: 712: 708: 704: 699: 697: 695: 690: 680: 676: 672: 668: 660: 656: 650:, dated 1303. 649: 645: 642:, appears in 641: 637: 633: 624: 622: 621:Pascal's rule 606: 603: 600: 597: 594: 574: 546: 542: 539: 536: 524: 512: 509: 506: 501: 498: 495: 483: 472: 469: 453: 439: 436: 424: 421: 375: 372: 343: 323: 310: 301: 286: 283: 280: 260: 257: 254: 245: 243: 239: 238:Blaise Pascal 235: 234:Western world 231: 227: 223: 219: 192: 188: 184: 180: 176: 172: 168: 164: 157: 153: 149: 145: 141: 137: 133: 126: 122: 118: 114: 110: 106: 99: 95: 91: 87: 83: 76: 72: 68: 64: 57: 53: 49: 42: 38: 31: 13243: 13233: 13212: 13203:Pascal's law 13134: 13126: 13113: 13091: 13059: 13055: 13049: 13024: 13020: 13014: 12989: 12985: 12979: 12951: 12941: 12924: 12920: 12907: 12893:(1): 12–13, 12890: 12886: 12880: 12870: 12863: 12854: 12844: 12819: 12815: 12809: 12665: 12661: 12651: 12623: 12613: 12585: 12575: 12555: 12548: 12523: 12519: 12512: 12503: 12496: 12487: 12479: 12446: 12442: 12436: 12403: 12397: 12391: 12375: 12369: 12363: 12338: 12332: 12328: 12322: 12297: 12291: 12285: 12274:. Retrieved 12270: 12261: 12235: 12229: 12220: 12205: 12185: 12178: 12168: 12136: 12132: 12122: 12106: 12101: 12084: 12078: 12058: 12051: 12039: 12011: 12004: 11996: 11992: 11946: 11926: 11919: 11911: 11906: 11873: 11867: 11747:Bean machine 11177: 10747: 10196:, where the 9944:Isaac Newton 9942: 9825: 9813: 9809: 9803: 9797: 9789: 9709: 9702: 9611: 9522: 9510: 9502: 9498: 9494: 9486: 9482: 9480: 9475: 9471: 9459: 9441: 9437: 9433: 9425: 9423: 9376: 9374: 9369: 9365: 9361: 9352: 9346: 9340: 9338: 9322: 9071: 8939: 8931: 8923: 8912: 8897: 8882: 8868: 8851: 8849: 8844: 8840: 8836: 8830: 8824: 8822: 8805: 8792:lookup table 8789: 8774: 8768:October 2016 8765: 8750:Please help 8738: 8685:dimensional 8664: 8652: 8108: 7981: 7977: 7968: 7574: 7283: 6952: 6676: 6673: 6663: 6657: 6653: 6649: 6645: 6638: 6634: 6627: 6623: 6618: 6613: 6609: 6604: 6598: 6594: 6578: 6574: 6569: 6565: 6560: 6556: 6554: 6545: 6374: 6369: 6366: 6362: 6360: 6046: 6023: 6017: 5982: 5836:prime number 5530: 4717:as follows: 4659: 4584: 4580: 4577:compositions 4559: 4544: 4440: 4319: 4223: 4010:items taken 3988:combinations 3985: 3982:Combinations 3870: 3621: 3535: 3524: 3377:is equal to 3266: 2620: 1615: 1549: 1128: 816: 774: 766: 753: 742:frontispiece 725: 723: 721:) in China. 710: 700: 693: 678: 675:Omar Khayyám 664: 640:rod numerals 454: 315: 246: 221: 215: 13185:Innovations 12927:(1): 1–15, 12664:: 1:31–33, 12341:: 145–148, 12238:(1): 1–17. 12109:, p. 2169. 11714:2.717181235 11687:1227 digits 10145:numeral in 9929:meromorphic 9800:dimensional 8843:edges, and 8813:tetrahedron 6591:tetrahedron 6583:dimensional 5985:: To count 218:mathematics 13335:Categories 12952:Symmetry 2 12624:Symmetry 2 12586:Symmetry 2 12276:2023-06-01 11914:, Vol. III 11847:References 11819:Proton NMR 9786:Extensions 9464:hypercubes 9440:(that is, 9357:orthogonal 9332:(called a 8647:See also: 6637:) = 1 and 5993:, convert 5334:Nilakantha 4618:equals to 4244:in row of 3654:, so that 734:Gersonides 726:Arithmetic 648:Zhu Shijie 13115:MathWorld 13098:EMS Press 12355:233356674 12314:218541210 12093:i27957284 12046:"Page 63" 11718:… 11681:⏞ 11678:… 11421:at entry 11263:− 11238:normalize 11195:− 10672:− 10663:− 10370:⋅ 10351:⋅ 10332:⋅ 10313:⋅ 10294:⋅ 10010:∞ 10007:→ 9900:Γ 9843:Γ 9771:… 9744:− 9735:− 9712:congruent 9657:⁡ 9566:⁡ 9507:real part 9334:hypercube 9039:− 9009:− 8998:− 8983:× 8818:simplices 8739:does not 8297:⁡ 7806:× 7752:× 7669:… 7539:× 7524:− 7513:− 7389:… 7251:× 7213:, 7183:× 7145:, 6926:− 6911:× 6896:− 6759:… 6514:− 6439:− 6421:∏ 6331:− 6295:∑ 6274:− 6246:∑ 6218:− 6198:− 6069:in order. 6031:Diagonals 5946:− 5901:− 5802:− 5753:− 5729:− 5689:− 5547:∑ 5380:− 5372:∞ 5357:∑ 5344:π 5319:π 5261:∞ 5258:→ 5200:≥ 5129:− 5118:⋅ 4998:∏ 4928:∏ 4828:− 4785:∏ 4739:∏ 4702:≥ 4691:for all 4498:… 4456:… 4406:. By the 4196:− 3779:− 3758:⋯ 3663:∑ 3439:− 3393:− 3196:− 3165:∑ 3078:− 3047:∑ 2973:∑ 2899:− 2871:∑ 2820:∑ 2798:− 2764:∑ 2716:∑ 2666:∑ 1995:∑ 1945:∑ 1760:∑ 1593:− 1403:− 1384:− 1370:… 1349:− 1317:− 1255:− 1217:∑ 767:Pascal's 758:Tartaglia 684:مثلث خیام 667:Al-Karaji 604:≤ 598:≤ 540:− 510:− 499:− 13310:Category 13284:(sister) 13276:(sister) 13268:(father) 13062:(9): 3, 13041:27957385 13006:27957164 12899:30213884 12836:27957091 11740:See also 11614:residues 11308:for all 11210:through 10742:yielding 10125:A radix 9892:, since 9710:If n is 9283: 84 9277: 14 9262: 64 9244: 60 9238: 12 9223: 32 9217: 80 9211: 80 9205: 40 9199: 10 9184: 16 9178: 32 9172: 24 9145: 12 8809:vertices 8796:polytope 8623:binomial 8610:counting 7040:, 7011:, 6982:, 6587:triangle 6024:Polarity 5830:, where 1135:binomial 707:Yang Hui 703:Jia Xian 696:th roots 636:Yang Hui 13320:Commons 13292:(niece) 13245:Pensées 13137:summary 13100:, 2001 12528:Bibcode 12471:1166003 12463:2324061 12428:0023257 12420:2304500 12378:: 247, 12252:2975209 12214:gallica 12017:Bibcode 11898:0028222 11890:2305028 11157:9699701 9703:is the 9643:Fourier 9552:Fourier 9493:of sin( 9453:√ 9446:√ 9372:-cube. 9271: 1 9232: 1 9193: 1 9166: 8 9160: 1 9151: 8 9139: 6 9133: 1 9124: 4 9118: 4 9112: 1 9103: 2 9097: 1 9088: 1 8926:squares 8760:removed 8745:sources 7961:fractal 7931:, etc. 6548:is the 6041:simplex 4654:in the 4651:A001142 627:History 401:choose 316:In the 304:Formula 228:of the 13258:Family 13249:(1669) 13188:Career 13039: 13004: 12966: 12897: 12834: 12638: 12600: 12563: 12469: 12461: 12426: 12418: 12353: 12312: 12300:: 51, 12250: 12193: 12113: 12091: 12066: 12027: 11934: 11896: 11888: 10198:degree 10149:(e.g. 8839:face, 8689:. The 8118:Plinko 6544:where 6001:. Let 5999:binary 5983:Parity 5194: 4347:, the 3536:Since 659:Pascal 13227:Works 13037:JSTOR 13002:JSTOR 12917:(PDF) 12895:JSTOR 12832:JSTOR 12459:JSTOR 12416:JSTOR 12351:S2CID 12310:S2CID 12248:JSTOR 12089:JSTOR 11886:JSTOR 11645:2.885 11481:equal 11288:with 11150:27433 10641:holds 10581:. If 10502:above 10279:14641 10250:) is 10158:14641 10042:limit 8856:cells 8679:(n-1) 6589:is a 5971:, so 5834:is a 3527:proof 1137:like 12964:ISBN 12636:ISBN 12598:ISBN 12561:ISBN 12191:ISBN 12111:ISBN 12064:ISBN 12025:ISBN 11932:ISBN 11706:1234 11637:1234 11632:1234 11601:1234 11579:1234 11574:1234 11359:> 11348:for 10833:and 10643:for 10624:< 10613:for 10222:term 9430:cube 9349:− 1) 9330:cube 8942:+ 1) 8934:+ 2) 8833:− 1) 8743:any 8741:cite 7069:and 6648:) = 6564:is: 4656:OEIS 4592:Rows 3529:(by 3456:and 3284:< 3278:< 1812:Now 719:楊輝三角 715:杨辉三角 13064:doi 13029:doi 12994:doi 12956:doi 12929:doi 12824:doi 12628:doi 12590:doi 12536:doi 12451:doi 12408:doi 12380:doi 12376:108 12343:doi 12331:", 12302:doi 12240:doi 12236:103 12212:at 12141:doi 11878:doi 11669:696 11663:977 11628:1.1 11570:1.1 11518:1.1 11479:is 11394:of 11119:164 11113:353 11107:560 11101:650 11095:560 11089:353 11083:164 10220:th 10000:lim 9949:row 9927:is 9808:or 9654:sin 9616:): 9563:sin 9304:128 9300:448 9296:672 9292:560 9288:280 9257:192 9253:240 9249:160 8820:). 8754:by 8294:exp 8260:21 8257:35 8254:35 8251:21 8234:15 8231:20 8228:15 8211:10 8208:10 6666:). 5997:to 5987:odd 5251:lim 4579:of 4552:.) 4414:as 728:of 216:In 13337:: 13112:. 13096:, 13090:, 13060:21 13058:, 13035:, 13025:59 13023:, 13000:, 12990:58 12988:, 12962:. 12950:. 12925:13 12923:, 12919:, 12891:13 12889:, 12853:, 12830:, 12820:57 12818:, 12783:11 12774:, 12756:11 12747:, 12729:11 12720:, 12702:11 12693:, 12675:11 12660:, 12634:. 12622:. 12596:. 12584:. 12534:, 12524:55 12522:, 12478:, 12467:MR 12465:, 12457:, 12447:99 12445:, 12424:MR 12422:, 12414:, 12404:54 12402:, 12374:, 12349:, 12339:96 12337:, 12308:, 12298:85 12296:, 12269:. 12246:. 12234:. 12155:^ 12137:16 12135:. 12131:. 12023:. 11894:MR 11892:, 11884:, 11874:56 11872:, 11855:^ 11722:10 11657:35 11456:11 11403:10 11296:01 11161:12 11142:12 11131:10 11125:56 11077:56 11071:10 11057:12 11052:12 11048:11 10992:11 10959:11 10860:11 10846:10 10842:11 10797:11 10558:11 10098:11 10016:11 9935:. 9818:. 9497:)/ 9474:)/ 9416:. 9404:81 8911:= 8907:+ 8896:= 8892:+ 8881:= 8877:+ 8867:= 8863:+ 8266:1 8263:7 8248:7 8245:1 8240:1 8237:6 8225:6 8222:1 8217:1 8214:5 8205:5 8202:1 8197:1 8194:4 8191:6 8188:4 8185:1 8180:1 8177:3 8174:3 8171:1 8166:1 8163:2 8160:1 8155:1 8152:1 8147:1 8101:20 8096:10 8075:10 7969:is 7824:21 7269:10 6552:. 5203:1. 4312:. 4300:35 1126:. 1079:, 1032:, 717:; 623:. 220:, 185:21 181:35 177:35 173:21 150:15 146:20 142:15 119:10 115:10 13167:e 13160:t 13153:v 13139:) 13118:. 13073:. 13066:: 13044:. 13031:: 13009:. 12996:: 12974:. 12972:. 12958:: 12936:. 12931:: 12902:. 12875:. 12858:. 12839:. 12826:: 12804:. 12787:4 12760:3 12733:2 12706:1 12679:0 12646:. 12644:. 12630:: 12608:. 12606:. 12592:: 12570:. 12543:. 12538:: 12530:: 12453:: 12410:: 12382:: 12358:. 12345:: 12329:e 12317:. 12304:: 12279:. 12254:. 12242:: 12200:. 12173:. 12149:. 12143:: 12117:. 12095:. 12072:. 12033:. 12019:: 11999:. 11997:n 11993:k 11975:) 11970:k 11967:n 11962:( 11940:. 11901:. 11880:: 11711:= 11702:1 11698:: 11695:0 11692:: 11672:: 11666:: 11660:: 11654:: 11651:2 11648:: 11642:= 11549:n 11527:n 11522:n 11497:1 11494:+ 11491:n 11465:n 11460:n 11435:1 11432:= 11429:k 11407:n 11382:1 11362:2 11356:n 11336:2 11316:n 11272:) 11266:1 11260:n 11256:n 11251:( 11224:1 11221:= 11218:k 11198:1 11192:n 11189:= 11186:k 11153:a 11147:= 11138:1 11134:: 11128:: 11122:: 11116:: 11110:: 11104:: 11098:: 11092:: 11086:: 11080:: 11074:: 11068:: 11065:1 11062:= 11024:n 11001:n 10996:n 10968:n 10963:n 10955:= 10950:n 10945:) 10939:n 10936:1 10931:+ 10928:1 10924:( 10917:n 10913:n 10892:n 10889:= 10886:a 10864:n 10856:= 10851:n 10819:n 10815:2 10811:= 10806:n 10801:1 10776:n 10756:a 10729:n 10709:c 10706:2 10701:d 10698:o 10695:m 10690:} 10687:) 10684:1 10681:+ 10678:c 10675:( 10669:, 10666:1 10660:c 10657:{ 10654:= 10651:a 10627:0 10621:c 10601:1 10598:+ 10595:a 10592:= 10589:c 10567:n 10562:a 10554:= 10549:n 10545:) 10541:1 10538:+ 10535:a 10532:( 10512:a 10488:a 10468:1 10465:= 10462:b 10442:b 10420:n 10416:) 10412:b 10409:+ 10406:a 10403:( 10392:. 10378:0 10374:a 10367:1 10364:+ 10359:1 10355:a 10348:4 10345:+ 10340:2 10336:a 10329:6 10326:+ 10321:3 10317:a 10310:4 10307:+ 10302:4 10298:a 10291:1 10288:= 10283:a 10258:i 10238:0 10235:= 10232:i 10208:i 10184:a 10162:a 10133:a 10107:n 10102:a 10094:= 10089:n 10085:) 10081:1 10078:+ 10075:a 10072:( 10052:n 10025:n 10020:a 10004:n 9978:a 9958:n 9915:) 9912:1 9909:+ 9906:z 9903:( 9879:C 9858:) 9855:1 9852:+ 9849:z 9846:( 9840:= 9837:! 9834:z 9768:, 9765:i 9762:+ 9759:, 9756:1 9753:+ 9750:, 9747:i 9741:, 9738:1 9732:, 9729:i 9726:+ 9716:i 9687:) 9682:] 9677:x 9671:1 9667:) 9663:x 9660:( 9648:[ 9638:( 9632:e 9629:R 9596:) 9591:] 9586:x 9580:5 9576:) 9572:x 9569:( 9557:[ 9547:( 9541:e 9538:R 9523:n 9511:n 9503:n 9499:x 9495:x 9487:x 9483:x 9476:x 9472:x 9460:V 9455:3 9448:2 9442:V 9438:V 9434:V 9426:n 9401:= 9396:4 9392:3 9377:m 9370:n 9366:n 9362:n 9353:n 9347:n 9345:( 9341:n 9057:. 9051:) 9046:k 9042:1 9036:n 9030:( 9024:+ 9018:) 9012:1 9006:k 9001:1 8995:n 8989:( 8980:2 8977:= 8971:) 8966:k 8963:n 8958:( 8940:x 8938:( 8932:x 8930:( 8913:4 8909:1 8905:3 8900:; 8898:6 8894:3 8890:3 8885:; 8883:4 8879:3 8875:1 8869:1 8865:1 8861:0 8845:3 8841:3 8837:1 8831:n 8829:( 8825:n 8807:( 8781:) 8775:( 8770:) 8766:( 8762:. 8748:. 8703:R 8683:n 8675:n 8619:= 8606:e 8596:) 8590:1 8585:4 8580:6 8575:4 8570:1 8563:. 8558:1 8553:3 8548:3 8543:1 8536:. 8531:. 8526:1 8521:2 8516:1 8509:. 8504:. 8499:. 8494:1 8489:1 8482:. 8477:. 8472:. 8467:. 8462:1 8456:( 8451:= 8442:) 8436:. 8431:4 8426:. 8421:. 8416:. 8409:. 8404:. 8399:3 8394:. 8389:. 8382:. 8377:. 8372:. 8367:2 8362:. 8355:. 8350:. 8345:. 8340:. 8335:1 8328:. 8323:. 8318:. 8313:. 8308:. 8302:( 8136:. 7982:n 7978:n 7915:) 7910:5 7907:7 7902:( 7895:, 7888:) 7883:5 7880:6 7875:( 7868:, 7861:) 7856:5 7853:5 7848:( 7821:= 7815:2 7812:7 7803:6 7800:= 7793:) 7788:2 7785:7 7780:( 7773:, 7770:6 7767:= 7761:1 7758:6 7749:1 7746:= 7739:) 7734:1 7731:6 7726:( 7719:, 7716:1 7713:= 7706:) 7701:0 7698:5 7693:( 7666:, 7660:3 7657:8 7651:, 7645:2 7642:7 7636:, 7630:1 7627:6 7600:) 7595:0 7592:5 7587:( 7560:. 7555:k 7551:k 7548:+ 7545:n 7533:) 7527:1 7521:k 7516:1 7510:k 7507:+ 7504:n 7498:( 7492:= 7486:) 7481:k 7477:k 7474:+ 7471:n 7465:( 7439:1 7436:= 7429:) 7424:0 7421:n 7416:( 7392:, 7386:, 7379:) 7374:2 7370:2 7367:+ 7364:n 7358:( 7351:, 7344:) 7339:1 7335:1 7332:+ 7329:n 7323:( 7316:, 7309:) 7304:0 7301:n 7296:( 7266:= 7260:2 7257:4 7248:5 7245:= 7238:) 7233:2 7230:5 7225:( 7201:5 7198:= 7192:1 7189:5 7180:1 7177:= 7170:) 7165:1 7162:5 7157:( 7133:1 7130:= 7123:) 7118:0 7115:5 7110:( 7083:5 7080:1 7054:4 7051:2 7025:3 7022:3 6996:2 6993:4 6967:1 6964:5 6938:. 6933:k 6929:k 6923:1 6920:+ 6917:n 6905:) 6899:1 6893:k 6889:n 6884:( 6878:= 6872:) 6867:k 6864:n 6859:( 6833:1 6830:= 6823:) 6818:0 6815:n 6810:( 6782:) 6777:n 6774:n 6769:( 6762:, 6756:, 6749:) 6744:1 6741:n 6736:( 6729:, 6722:) 6717:0 6714:n 6709:( 6685:n 6664:x 6662:( 6658:d 6654:P 6650:x 6646:x 6644:( 6642:1 6639:P 6635:x 6633:( 6631:0 6628:P 6624:x 6622:( 6619:d 6614:x 6610:x 6608:( 6605:d 6599:d 6595:P 6581:- 6579:d 6575:d 6570:d 6566:P 6561:d 6557:P 6546:n 6532:, 6526:) 6521:d 6517:1 6511:d 6508:+ 6505:n 6499:( 6493:= 6487:! 6484:d 6478:) 6475:d 6472:( 6468:n 6462:= 6459:) 6456:k 6453:+ 6450:n 6447:( 6442:1 6436:d 6431:0 6428:= 6425:k 6414:! 6411:d 6407:1 6402:= 6399:) 6396:n 6393:( 6388:d 6384:P 6370:n 6367:d 6363:n 6340:. 6337:) 6334:1 6328:n 6325:( 6320:i 6316:P 6310:d 6305:0 6302:= 6299:i 6291:= 6288:) 6285:i 6282:( 6277:1 6271:d 6267:P 6261:n 6256:0 6253:= 6250:i 6242:= 6232:) 6229:n 6226:( 6221:1 6215:d 6211:P 6207:+ 6204:) 6201:1 6195:n 6192:( 6187:d 6183:P 6179:= 6172:) 6169:n 6166:( 6161:d 6157:P 6149:, 6146:1 6143:= 6140:) 6137:0 6134:( 6129:d 6125:P 6121:= 6114:) 6111:n 6108:( 6103:0 6099:P 6080:. 6018:n 6013:. 6007:2 6003:x 5995:n 5991:n 5979:. 5977:p 5973:p 5969:p 5955:! 5952:) 5949:k 5943:p 5940:( 5937:! 5934:k 5910:! 5907:) 5904:k 5898:p 5895:( 5892:! 5889:k 5884:! 5881:p 5874:= 5867:) 5862:k 5859:p 5854:( 5840:p 5832:p 5828:p 5823:. 5811:5 5808:= 5805:1 5799:6 5796:= 5791:3 5787:C 5762:) 5756:2 5750:m 5745:m 5742:2 5736:( 5722:) 5717:m 5713:m 5710:2 5704:( 5697:= 5692:1 5686:m 5682:C 5657:m 5654:2 5651:= 5648:n 5626:. 5620:) 5615:n 5611:n 5608:2 5602:( 5596:= 5591:2 5584:) 5579:k 5576:n 5571:( 5562:n 5557:0 5554:= 5551:k 5531:n 5529:2 5525:n 5520:. 5496:) 5491:2 5487:2 5484:+ 5481:n 5478:2 5472:( 5463:) 5458:2 5454:1 5451:+ 5448:n 5445:2 5439:( 5430:) 5425:1 5421:1 5418:+ 5415:n 5412:2 5406:( 5397:1 5394:+ 5391:n 5387:) 5383:1 5377:( 5367:1 5364:= 5361:n 5353:+ 5350:3 5347:= 5308:. 5294:n 5289:) 5283:n 5280:1 5275:+ 5272:1 5268:( 5255:n 5247:= 5244:e 5224:e 5197:n 5191:, 5186:n 5181:) 5176:n 5172:1 5169:+ 5166:n 5160:( 5155:= 5148:2 5143:n 5139:s 5132:1 5126:n 5122:s 5113:1 5110:+ 5107:n 5103:s 5076:! 5073:n 5066:n 5062:) 5058:1 5055:+ 5052:n 5049:( 5043:= 5032:2 5028:! 5024:k 5020:1 5013:n 5008:0 5005:= 5002:k 4992:1 4989:+ 4986:n 4982:! 4978:n 4968:2 4964:! 4960:k 4956:1 4949:1 4946:+ 4943:n 4938:0 4935:= 4932:k 4922:2 4919:+ 4916:n 4912:! 4908:) 4905:1 4902:+ 4899:n 4896:( 4890:= 4883:n 4879:s 4873:1 4870:+ 4867:n 4863:s 4837:! 4834:) 4831:k 4825:n 4822:( 4819:! 4816:k 4811:! 4808:n 4800:n 4795:0 4792:= 4789:k 4781:= 4775:) 4770:k 4767:n 4762:( 4754:n 4749:0 4746:= 4743:k 4735:= 4730:n 4726:s 4705:0 4699:n 4677:n 4673:s 4661:e 4645:. 4631:n 4627:2 4606:n 4585:k 4581:n 4545:n 4527:1 4524:+ 4521:x 4501:} 4495:, 4492:0 4489:, 4486:0 4483:, 4480:1 4477:, 4474:1 4471:, 4468:0 4465:, 4462:0 4459:, 4453:{ 4422:n 4391:2 4388:1 4382:= 4379:p 4355:n 4333:n 4329:2 4297:= 4290:) 4285:3 4282:7 4277:( 4252:n 4232:k 4220:. 4205:! 4202:) 4199:k 4193:n 4190:( 4187:! 4184:k 4179:! 4176:n 4170:= 4164:) 4159:k 4156:n 4151:( 4145:= 4139:k 4135:C 4129:n 4121:= 4116:n 4111:k 4106:C 4101:= 4098:) 4095:k 4092:, 4089:n 4086:( 4082:C 4058:n 4038:k 4018:k 3998:n 3966:n 3944:n 3940:2 3919:n 3899:n 3879:n 3857:. 3852:n 3848:2 3844:= 3839:n 3835:) 3831:1 3828:+ 3825:1 3822:( 3819:= 3813:) 3808:n 3805:n 3800:( 3794:+ 3788:) 3782:1 3776:n 3772:n 3767:( 3761:+ 3755:+ 3749:) 3744:1 3741:n 3736:( 3730:+ 3724:) 3719:0 3716:n 3711:( 3705:= 3699:) 3694:k 3691:n 3686:( 3678:n 3673:0 3670:= 3667:k 3642:1 3639:= 3636:y 3633:= 3630:x 3605:n 3601:) 3597:1 3594:+ 3588:b 3585:a 3579:( 3574:n 3570:b 3566:= 3561:n 3557:) 3553:b 3550:+ 3547:a 3544:( 3508:n 3504:) 3500:1 3497:+ 3494:x 3491:( 3469:k 3465:x 3442:1 3436:k 3432:x 3409:k 3405:a 3401:+ 3396:1 3390:k 3386:a 3363:1 3360:+ 3357:n 3353:) 3349:1 3346:+ 3343:x 3340:( 3318:k 3314:x 3293:1 3290:+ 3287:n 3281:k 3275:0 3249:. 3244:1 3241:+ 3238:n 3234:x 3230:+ 3225:k 3221:x 3217:) 3212:k 3208:a 3204:+ 3199:1 3193:k 3189:a 3185:( 3180:n 3175:1 3172:= 3169:k 3161:+ 3156:0 3152:x 3148:= 3136:1 3133:+ 3130:n 3126:x 3120:n 3116:a 3112:+ 3107:k 3103:x 3099:) 3094:k 3090:a 3086:+ 3081:1 3075:k 3071:a 3067:( 3062:n 3057:1 3054:= 3051:k 3043:+ 3038:0 3034:x 3028:0 3024:a 3020:= 3008:k 3004:x 2998:k 2994:a 2988:n 2983:1 2980:= 2977:k 2969:+ 2964:0 2960:x 2954:0 2950:a 2946:+ 2941:1 2938:+ 2935:n 2931:x 2925:n 2921:a 2917:+ 2912:k 2908:x 2902:1 2896:k 2892:a 2886:n 2881:1 2878:= 2875:k 2867:= 2855:k 2851:x 2845:k 2841:a 2835:n 2830:0 2827:= 2824:k 2816:+ 2811:k 2807:x 2801:1 2795:k 2791:a 2785:1 2782:+ 2779:n 2774:1 2771:= 2768:k 2760:= 2751:k 2747:x 2741:k 2737:a 2731:n 2726:0 2723:= 2720:k 2712:+ 2707:1 2704:+ 2701:i 2697:x 2691:i 2687:a 2681:n 2676:0 2673:= 2670:i 2641:1 2638:+ 2635:i 2632:= 2629:k 2594:) 2589:5 2586:5 2581:( 2569:) 2564:4 2561:5 2556:( 2544:) 2539:3 2536:5 2531:( 2519:) 2514:2 2511:5 2506:( 2494:) 2489:1 2486:5 2481:( 2469:) 2464:0 2461:5 2456:( 2441:) 2436:4 2433:4 2428:( 2416:) 2411:3 2408:4 2403:( 2391:) 2386:2 2383:4 2378:( 2366:) 2361:1 2358:4 2353:( 2341:) 2336:0 2333:4 2328:( 2313:) 2308:3 2305:3 2300:( 2288:) 2283:2 2280:3 2275:( 2263:) 2258:1 2255:3 2250:( 2238:) 2233:0 2230:3 2225:( 2210:) 2205:2 2202:2 2197:( 2185:) 2180:1 2177:2 2172:( 2160:) 2155:0 2152:2 2147:( 2132:) 2127:1 2124:1 2119:( 2107:) 2102:0 2099:1 2094:( 2079:) 2074:0 2071:0 2066:( 2035:. 2030:k 2026:x 2020:k 2016:a 2010:n 2005:0 2002:= 1999:k 1991:+ 1986:1 1983:+ 1980:i 1976:x 1970:i 1966:a 1960:n 1955:0 1952:= 1949:i 1941:= 1936:n 1932:) 1928:1 1925:+ 1922:x 1919:( 1916:+ 1911:n 1907:) 1903:1 1900:+ 1897:x 1894:( 1891:x 1888:= 1883:n 1879:) 1875:1 1872:+ 1869:x 1866:( 1863:) 1860:1 1857:+ 1854:x 1851:( 1848:= 1843:1 1840:+ 1837:n 1833:) 1829:1 1826:+ 1823:x 1820:( 1800:. 1795:k 1791:x 1785:k 1781:a 1775:n 1770:0 1767:= 1764:k 1756:= 1751:n 1747:) 1743:1 1740:+ 1737:x 1734:( 1714:1 1711:= 1708:y 1686:n 1682:) 1678:1 1675:+ 1672:x 1669:( 1647:1 1644:+ 1641:n 1637:) 1633:y 1630:+ 1627:x 1624:( 1601:y 1596:1 1590:n 1586:x 1563:n 1559:x 1536:. 1530:) 1525:k 1522:n 1517:( 1511:= 1506:k 1502:a 1481:n 1459:k 1455:a 1434:, 1429:n 1425:y 1419:n 1415:a 1411:+ 1406:1 1400:n 1396:y 1392:x 1387:1 1381:n 1377:a 1373:+ 1367:+ 1362:2 1358:y 1352:2 1346:n 1342:x 1336:2 1332:a 1328:+ 1325:y 1320:1 1314:n 1310:x 1304:1 1300:a 1296:+ 1291:n 1287:x 1281:0 1277:a 1273:= 1268:k 1264:y 1258:k 1252:n 1248:x 1242:k 1238:a 1232:n 1227:0 1224:= 1221:k 1213:= 1208:n 1204:) 1200:y 1197:+ 1194:x 1191:( 1171:n 1151:y 1148:+ 1145:x 1114:1 1111:= 1104:) 1099:2 1096:2 1091:( 1067:2 1064:= 1057:) 1052:1 1049:2 1044:( 1020:1 1017:= 1010:) 1005:0 1002:2 997:( 973:, 968:2 964:y 958:0 954:x 949:1 945:+ 940:1 936:y 930:1 926:x 921:2 917:+ 912:0 908:y 902:2 898:x 893:1 889:= 884:2 880:y 876:+ 873:y 870:x 867:2 864:+ 859:2 855:x 851:= 846:2 842:) 838:y 835:+ 832:x 829:( 773:( 713:( 694:n 681:( 607:n 601:k 595:0 575:n 552:) 547:k 543:1 537:n 531:( 525:+ 519:) 513:1 507:k 502:1 496:n 490:( 484:= 478:) 473:k 470:n 465:( 440:1 437:= 430:) 425:0 422:0 417:( 403:k 399:n 381:) 376:k 373:n 368:( 344:k 324:n 287:0 284:= 281:k 261:0 258:= 255:n 193:1 189:7 169:7 165:1 158:1 154:6 138:6 134:1 127:1 123:5 111:5 107:1 100:1 96:4 92:6 88:4 84:1 77:1 73:3 69:3 65:1 58:1 54:2 50:1 43:1 39:1 32:1

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