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
9323:
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
4264:
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
8815:
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
9359:
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
1186:
7402:
300:
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,
4511:
10037:
6378:
5094:
11989:
4567:
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.
7287:
457:
8673:
rather than matrices. Recognising the geometric operations, such as rotations, allows the algebra operations to be discovered. Just as each row,
7457:
6700:
12059:
From
Alexandria, Through Baghdad: Surveys and Studies in the Ancient Greek and Medieval Islamic Mathematical Sciences in Honor of J.L. Berggren
9790:
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.
9072:
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:
8950:
7619:
6851:
6048:
13136:
9714:
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.
4445:
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.
9721:
3539:
748:
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.
701:
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:
8755:
8653:
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.),
9895:
9873:
8697:
5517:
11757:
11512:
8908:
8823:
To understand why this pattern exists, one must first understand that the process of building an
8744:
7959:
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
12012:
Encyclopaedia of the
History of Science, Technology, and Medicine in Non-Western Cultures
11829:
11813:
11808:
11777:
11486:
11424:
11213:
10881:
10457:
10227:
10146:
9814:
9804:
9711:
7964:
6010:
5333:
4516:
4411:
3911:
th power of 2. This is equivalent to the statement that the number of subsets of an
3622:
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
8878:
8855:
8654:
7938:
6680:
6073:
6005:
be the number of 1s in the binary representation. Then the number of odd terms will be
5975:
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.
698:
based on the binomial expansion, and therefore on the binomial coefficients.
658:
639:
244:
studied it centuries before him in Persia, India, China, Germany, and Italy.
241:
237:
233:
12850:
3267:
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
736:
during the early 14th century, using the multiplicative formula for them.
308:
13032:
12997:
12871:
Finding any row of Pascal's triangle extending the concept of power of 11
12827:
12092:
11178:
with compound digits (delimited by ":") in radix twelve. The digits from
8812:
6590:
6582:
3987:
985:
the coefficients are the entries in the second row of Pascal's triangle:
673:
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
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12302:doi
12240:doi
12236:103
12212:at
12141:doi
11878:doi
11669:696
11663:977
11628:1.1
11570:1.1
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10000:lim
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8863:+
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11967:n
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11527:n
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11429:k
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10811:=
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9460:V
9455:3
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9438:V
9434:V
9426:n
9401:=
9396:4
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8980:2
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8971:)
8966:k
8963:n
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8932:x
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8898:6
8894:3
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8879:3
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8829:(
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8781:)
8775:(
8770:)
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8762:.
8748:.
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8675:n
8619:=
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5176:n
5172:1
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5160:(
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4101:=
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3844:=
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3822:(
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3508:n
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2160:)
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2066:(
2035:.
2030:k
2026:x
2020:k
2016:a
2010:n
2005:0
2002:=
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1991:+
1986:1
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1980:i
1976:x
1970:i
1966:a
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1955:0
1952:=
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1941:=
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1922:x
1919:(
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1911:n
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1903:1
1900:+
1897:x
1894:(
1891:x
1888:=
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1872:+
1869:x
1866:(
1863:)
1860:1
1857:+
1854:x
1851:(
1848:=
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1840:+
1837:n
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1829:1
1826:+
1823:x
1820:(
1800:.
1795:k
1791:x
1785:k
1781:a
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1740:+
1737:x
1734:(
1714:1
1711:=
1708:y
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1682:)
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1672:x
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1641:n
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1563:n
1559:x
1536:.
1530:)
1525:k
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1517:(
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1459:k
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1429:n
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1310:x
1304:1
1300:a
1296:+
1291:n
1287:x
1281:0
1277:a
1273:=
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1252:n
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1242:k
1238:a
1232:n
1227:0
1224:=
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1213:=
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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:=
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1052:1
1049:2
1044:(
1020:1
1017:=
1010:)
1005:0
1002:2
997:(
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968:2
964:y
958:0
954:x
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945:+
940:1
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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:(
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681:(
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575:n
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547:k
543:1
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525:+
519:)
513:1
507:k
502:1
496:n
490:(
484:=
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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:=
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261:0
258:=
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193:1
189:7
169:7
165:1
158:1
154:6
138:6
134:1
127:1
123:5
111:5
107:1
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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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