A014410 -id:A014410 - cp's OEIS frontend

A075780 Triangle T(n,k) = f(n,k,n-2), n >= 2, 1 <= k <= n-1, where f is given below.

0, 3, 3, 12, 14, 12, 30, 45, 45, 30, 60, 114, 138, 114, 60, 105, 245, 357, 357, 245, 105, 168, 468, 808, 960, 808, 468, 168, 252, 819, 1647, 2286, 2286, 1647, 819, 252, 360, 1340, 3090, 4935, 5740, 4935, 3090, 1340, 360, 495, 2079, 5423, 9834, 13090, 13090, 9834, 5423

Offset: 2

N. J. A. Sloane, Oct 17 2002

Michel Lassalle, A new family of positive integers

Cf. A014410 and A007318 for f(n, k, n), A075779 and A075798 for f(n, k, n-1) and A075780 and A075837 for f(n, k, n-2).

f := proc(n,p,k) convert( binomial(n,k)*hypergeom([1-k,-p,p-n],[1-n,1],1), `StandardFunctions`); end;

t[n_, k_] := n*(n-1)/2*HypergeometricPFQ[{-k, 3-n, k-n}, {1, 1-n}, 1]; Table[t[n, k], {n, 2, 12}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

f(n, p, k) = binomial(n, k)*hypergeom([1-k, -p, p-n], [1-n, 1], 1).

A075798 Triangle T(n,k) = f(n,k,n-1), n >= 0, 0 <= k <= n, where f is given below.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 12, 16, 12, 1, 1, 20, 35, 35, 20, 1, 1, 30, 66, 84, 66, 30, 1, 1, 42, 112, 175, 175, 112, 42, 1, 1, 56, 176, 328, 400, 328, 176, 56, 1, 1, 72, 261, 567, 819, 819, 567, 261, 72, 1, 1, 90, 370, 920, 1540, 1820, 1540, 920, 370, 90, 1, 1, 110, 506, 1419, 2706, 3696, 3696, 2706, 1419, 506, 110, 1, 1, 132, 672

Offset: 1

N. J. A. Sloane, Oct 17 2002

			1; 1,1; 1,2,1; 1,6,6,1; ...

Michel Lassalle, A new family of positive integers

Cf. A014410 and A007318 for f(n, k, n), A075779 and A075798 for f(n, k, n-1) and A075780 and A075837 for f(n, k, n-2).

f := proc(n,p,k) convert( binomial(n,k)*hypergeom([1-k,-p,p-n],[1-n,1],1), `StandardFunctions`); end;

f[n_, p_, k_] := Binomial[n, k]*HypergeometricPFQ[{1 - k, -p, p-n}, {1-n, 1}, 1]; t[n_, n_] = t[, 0] = 1; t[n, k_] := f[n, k, n-1]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 17 2014 *)

f(n, p, k) = binomial(n, k)*hypergeom([1-k, -p, p-n], [1-n, 1], 1).

A075837 Triangle T(n,k) = f(n,k,n-2), n >= 0, 0 <= k <= n, where f is given below.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 3, 3, 1, 1, 12, 14, 12, 1, 1, 30, 45, 45, 30, 1, 1, 60, 114, 138, 114, 60, 1, 1, 105, 245, 357, 357, 245, 105, 1, 1, 168, 468, 808, 960, 808, 468, 168, 1, 1, 252, 819, 1647, 2286, 2286, 1647, 819, 252, 1, 1, 360, 1340, 3090, 4935, 5740, 4935, 3090, 1340, 360, 1, 1, 495, 2079, 5423, 9834, 13090, 13090, 9834, 5423

Offset: 1

N. J. A. Sloane, Oct 17 2002

			1; 1,1; 1,0,1; 1,3,3,1; ...

Michel Lassalle, A new family of positive integers

Cf. A014410 and A007318 for f(n, k, n), A075779 and A075798 for f(n, k, n-1) and A075780 and A075837 for f(n, k, n-2).

f := proc(n,p,k) convert( binomial(n,k)*hypergeom([1-k,-p,p-n],[1-n,1],1), `StandardFunctions`); end;

f[n_, p_, k_] := Binomial[n, k]*HypergeometricPFQ[{1 - k, -p, p-n}, {1-n, 1}, 1]; t[n_, n_] = t[, 0] = 1; t[n, k_] := f[n, k, n-2]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 17 2014 *)

f(n, p, k) = binomial(n, k)*hypergeom([1-k, -p, p-n], [1-n, 1], 1).

A257241 Irregular triangle read by rows: Stifel's version of the arithmetical triangle.

Original entry on oeis.org

1, 2, 3, 3, 4, 6, 5, 10, 10, 6, 15, 20, 7, 21, 35, 35, 8, 28, 56, 70, 9, 36, 84, 126, 126, 10, 45, 120, 210, 252, 11, 55, 165, 330, 462, 462, 12, 66, 220, 495, 792, 924, 13, 78, 286, 715, 1287, 1716, 1716, 14, 91, 364, 1001, 2002, 3003, 3432, 15, 105, 455, 1365, 3003, 5005, 6435, 6435

Offset: 1

Wolfdieter Lang, May 22 2015

The row length of this array is A008619(n-1), for n >= 1: 1, 1, 2, 2, ...
This is a truncated version of Pascal's triangle used by Michael Stifel (1487?-1567). It already appeared on the title page (frontispiece) of Peter Apianus's book of 1527 on business arithmetic: "Eyn Newe Und wolgegründte underweysung aller Kauffmanns Rechnung in dreyen Büchern". See the Kac reference, p. 394 and Table 12.1 on p. 395. It appeared in Stifel's 1553 edition of Rudolff's Coß: "Die Coß Christoffs Rudolffs. Die schönen Exemplen der Coß Durch Michael Stifel gebessert und sehr gemehrt." See the MacTutor Archive link and the Alten reference.
The row sums give A258143. The alternating row sums give A258144.
T(n,A008619(n-1)) = A001405(n). - Reinhard Zumkeller, May 22 2015

			The irregular triangle T(n, m) begins:
  n\m|  1    2    3    4    5    6    7 ...
  ---+-------------------------------------
   1 |  1
   2 |  2
   3 |  3    3
   4 |  4    6
   5 |  5   10   10
   6 |  6   15   20
   7 |  7   21   35   35
   8 |  8   28   56   70
   9 |  9   36   84  126  126
  10 | 10   45  120  210  252
  11 | 11   55  165  330  462  462
  12 | 12   66  220  495  792  924
  13 | 13   78  286  715 1287 1716 1716
  ...

H.-W. Alten et al., 4000 Jahre Algebra, 2. Auflage, Springer, 2014, p. 260.
Victor J. Kac, A History of Mathematics, third edition, Addison-Wesley, 2009.
Reich, Karin; Michael Stifel. In: Folkerts, Menso; Eberhard Knobloch; Karin Reich: Maß, Zahl und Gewicht: Mathematik als Schlüssel zu Weltverständnis und Weltbeherrschung. Wolfenbüttel 1989, S. 73 - 95 und 373.

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
MacTutor History of Mathematics archive, Petrus Apianus.
MacTutor History of Mathematics archive, Michael Stifel
Maurer, Bertram, Michael Stifel, 1999, Kolping-Kolleg Stuttgart.
Wikipedia, Petrus Apianus.
Wikipedia, Michael Stifel.

Cf. A007318, A258143, A258144, A014410 (Scheubel's version).

Cf. A001405 (right edge).

a257241 n k = a257241_tabf !! (n-1) !! (k-1)
a257241_row n = a257241_tabf !! (n-1)
a257241_tabf = iterate stifel [1] where
   stifel xs@(x:_) = if odd x then xs' else xs' ++ [last xs']
                     where xs' = zipWith (+) xs (1 : xs)
-- Reinhard Zumkeller, May 22 2015

Table[Binomial[n, m], {n, 15}, {m, Ceiling[n/2]}] (* Paolo Xausa, Nov 14 2024 *)

T(n, m) = binomial(n, m), n >= 1, m = 1, 2, ..., ceiling(n/2).

O.g.f. row m = 1, 2, ..., 4 (with leading zeros): x/(1-x)^2, x^3*(3-3*x+x^2)/(1-x)^3, x^5*(10-20*x+15*x^2-4*x^3)/(1-x)^4, x^7*(35-105*x+126*x^2-70*x^3+15*x^4)/(1-x)^5.

A259052 Sum of Pascal triples.

Original entry on oeis.org

3, 4, 4, 4, 5, 6, 8, 6, 5, 6, 8, 13, 12, 13, 8, 6, 7, 10, 19, 20, 26, 20, 19, 10, 7, 8, 12, 26, 30, 45, 40, 45, 30, 26, 12, 8, 9, 14, 34, 42, 71, 70, 90, 70, 71, 42, 34, 14, 9, 10, 16, 43, 56, 105, 112, 161, 140, 161, 112, 105, 56, 43, 16, 10

Offset: 1

Wolfdieter Lang, Jun 27 2015

The sequence of row lengths of this irregular triangle is A005408(n-1) = 2*n - 1.
This entry is motivated by A258445 from Craig Knecht. There the minima of the Pascal triples are given.
A Pascal triple PT(n, k) for n >= 1, k = 1, 2, ..., 2*n-1 is defined for even k by (P(n-1, k/2-1), P(n-1, k/2), P(n, k/2)) with P(n, k) = A007318(n, k) = binomial(n, k), and for odd k by (P(n-1, (k-1)/2), P(n, (k-1)/2), P(n, (k+1)/2)).
The strip S_n between row n-1 and n of Pascal's triangle (written as symmetric equilateral triangle) is divided into 2*n-1 small equilateral up - down triangles connecting neighboring entries of Pascal's triangle. For odd k these triangles have their base on row n of Pascal's triangle (up triangles), and for even n their base is on row n-1 (down triangles). There are n up triangles and n-1 down triangles in strip S_n.
The present irregular triangle gives the sum of the Pascal triples.
This is motivated by the idea (see A258445) of considering equal touching cylinders (closed only with a bottom disk) with centers at the corners of the small up and down triangles and radius r/2 if the side of each triangle has length r. They are filled with a liquid to a height h with h/r given by the Pascal entry at the center of the bottom of the cylinder. If, for each of the three pairs from a triple of touching cylinders a hole on the bottom of the vertical touching line is opened, then the new height H of the liquid for such a triple will be the arithmetic mean of the three original heights of the three touching cylinders. The ratio H/r will be 1/3 of the corresponding irregular triangle entry for this Pascal triple.
The row sums of this irregular triangle give 3*A033484(n-1), n >= 1.

			The irregular triangle T(n, k) starts:
  n\k 1  2  3  4  5  6  7  8  9 10 11 12 13 ...
  1:  3
  2:  4  4  4
  3:  5  6  8  6  5
  4:  6  8 13 12 13  8  6
  5:  7 10 19 20 26 20 19 10  7
  6:  8 12 26 30 45 40 45 30 26 12  8
  7:  9 14 34 42 71 70 90 70 71 42 34 14  9
  ...
T(3, 2) = 6 from the sum of the Pascal triple (1, 2, 3) (from the first down triangle in Pascal's triangle strip S_3).
The height ratio H/r for this Pascal triple PT(3, 2) = (1, 2, 3) is (1 + 2 + 3)/3 = T(3, 2)/3 = 2.

Craig Knecht, Pascal triples
Craig Knecht, Sum of triples table

Cf. A007318, A258445, A005408, 3*A033484.

tabl(nn) = {for (n=1, nn, for (k=1, 2*n-1, kk = (k+1)\2; if (k%2, v = binomial(n-1, kk-1) + binomial(n, kk-1) + binomial(n, kk), v = binomial(n, kk) + binomial(n-1, kk-1) + binomial(n-1, kk)); print1(v, ", ");); print(););} \\ Michel Marcus, Jun 27 2015

T(n, 2*m) = P(n-1, m-1) + P(n-1, m) + P(n, m) with P(n, k) = A007318(n, k) = binomial(n, k), for m = 1..n-1, and
T(n, 2*m-1) = P(n-1, m-1) + P(n, m-1) + P(n, m) for m = 1..n.
T(n, 2*m-1) = A028263(n-1, m-1), n >= 1, m = 1..n, and T(n, 2*m) = 2*A014410(n, m), n >= 2, m = 1..n-1.

A091963 a(n) is the smallest gcd of two interior numbers on row n of Pascal's triangle ("interior" means that the 1's at the ends of the rows are excluded).

Original entry on oeis.org

2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 3, 13, 2, 3, 2, 17, 2, 19, 4, 3, 2, 23, 3, 5, 2, 3, 4, 29, 6, 31, 2, 3, 2, 5, 4, 37, 2, 3, 5, 41, 6, 43, 4, 3, 2, 47, 3, 7, 2, 3, 4, 53, 2, 5, 7, 3, 2, 59, 4, 61, 2, 7, 2, 5, 6, 67, 4, 3, 10, 71, 4, 73, 2, 3, 4, 7, 2, 79, 5, 3, 2, 83, 12, 5, 2, 3, 4, 89, 9, 7, 4, 3, 2, 5, 3

Offset: 2

David Wasserman, Mar 13 2004

nonn

The reference contains a simple proof that there are no 1's in this sequence.

			In row 8, the interior numbers 8, 28, 56 and 70; gcd(8, 28) = 4; gcd(8, 56) = 8; gcd(8, 70) = 2; gcd(28, 56) = 28; gcd(28, 70) = 14; gcd(56, 70) = 14. The smallest of these is 2, so a(8) = 2.

R. K. Guy, Unsolved Problems in Number Theory, Sections B31, B33.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A014410.

seq(min(seq(igcd(n,binomial(n,k)),k=1..floor(n/2))), n=2..100); # Robert Israel, Jun 17 2014

a(n) = {v = vector(n\2, i, binomial(n, i)); mgcd = n; for (i=1, #v, for (j=i+1, #v, mgcd = min(gcd(v[i], v[j]), mgcd););); return (mgcd);} \\ Michel Marcus, Jun 16 2013

A180986 T(n,k) = number of n X k binary matrices with rows in lexicographically nondecreasing order and columns in lexicographically nonincreasing order.

Original entry on oeis.org

2, 3, 3, 4, 6, 4, 5, 10, 10, 5, 6, 15, 20, 15, 6, 7, 21, 35, 35, 21, 7, 8, 28, 56, 70, 56, 28, 8, 9, 36, 84, 126, 126, 84, 36, 9, 10, 45, 120, 210, 252, 210, 120, 45, 10, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 13, 78, 286, 715

Offset: 1

R. H. Hardin, Sep 30 2010

Table starts:
..2..3...4....5....6....7.....8.....9....10.....11.....12.....13......14
..3..6..10...15...21...28....36....45....55.....66.....78.....91.....105
..4.10..20...35...56...84...120...165...220....286....364....455.....560
..5.15..35...70..126..210...330...495...715...1001...1365...1820....2380
..6.21..56..126..252..462...792..1287..2002...3003...4368...6188....8568
..7.28..84..210..462..924..1716..3003..5005...8008..12376..18564...27132
..8.36.120..330..792.1716..3432..6435.11440..19448..31824..50388...77520
..9.45.165..495.1287.3003..6435.12870.24310..43758..75582.125970..203490
.10.55.220..715.2002.5005.11440.24310.48620..92378.167960.293930..497420
.11.66.286.1001.3003.8008.19448.43758.92378.184756.352716.646646.1144066
Is this (apart from offsets and formatting) the same sequence as A014410? [R. J. Mathar, Oct 02 2010]
Yes, because it obeys the recursion formula for binomial coefficients: the top left element is either 0 (leaving T(n-1,k) ways to fill the rest) or 1 (leaving T(n,k-1) ways to fill the rest). [Karl W. Heuer, Aug 25 2014]

			All solutions for 3 X 3:
..0..0..0....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0
..0..0..0....0..0..0....0..0..0....1..0..0....1..0..0....1..1..0....1..0..0
..1..0..0....1..1..0....1..1..1....1..0..0....1..1..0....1..1..0....1..1..1
...
..0..0..0....0..0..0....1..0..0....1..0..0....1..0..0....1..0..0....1..0..0
..1..1..0....1..1..1....1..0..0....1..0..0....1..1..0....1..0..0....1..1..0
..1..1..1....1..1..1....1..0..0....1..1..0....1..1..0....1..1..1....1..1..1
...
..1..0..0....1..1..0....1..1..0....1..1..0....1..1..1....0..0..0
..1..1..1....1..1..0....1..1..0....1..1..1....1..1..1....0..0..0
..1..1..1....1..1..0....1..1..1....1..1..1....1..1..1....0..0..0

R. H. Hardin, Table of n, a(n) for n=1..544

A014410 is the same sequence viewed as a triangle.

A014719 Squares of elements in Pascal triangle (by row) that are not 1.

Original entry on oeis.org

4, 9, 9, 16, 36, 16, 25, 100, 100, 25, 36, 225, 400, 225, 36, 49, 441, 1225, 1225, 441, 49, 64, 784, 3136, 4900, 3136, 784, 64, 81, 1296, 7056, 15876, 15876, 7056, 1296, 81, 100, 2025, 14400, 44100, 63504, 44100, 14400, 2025, 100, 121, 3025

Offset: 2

Mohammad K. Azarian

Number of ways to place 2k*(n-k) nonattacking ferses on a 2k X (2n-2k) board. - Tricia Muldoon Brown, Dec 12 2018

			Triangle begins:
  4,
  9, 9,
  16, 36, 16,
  25, 100, 100, 25,
  36, 225, 400, 225, 36,
  ...

Vincenzo Librandi, Table of n, a(n) for n = 2..5461
Tricia Muldoon Brown, The Problem of Pawns, arXiv:1811.09606 [math.CO], 2018.

Cf. A007318, A014410.

T:=(n,k)->binomial(n,k)^2: seq(seq(T(n,k),k=1..n-1),n=2..11); # Muniru A Asiru, Dec 12 2018

Select[Flatten[Table[Binomial[n, i]^2, {n, 0, 25}, {i, 0, n}]], #>1 &] (* Vincenzo Librandi, Nov 19 2018 *)

a(n) = A014410(n)^2. - Sean A. Irvine, Nov 18 2018

More terms from Sean A. Irvine, Nov 18 2018

Offset corrected by Joerg Arndt, Nov 19 2018

A140414 Triangle T(p,s) showing the coefficients of sequences which are half their p-th differences.

Original entry on oeis.org

3, 2, 1, 3, -3, 3, 4, -6, 4, 1, 5, -10, 10, -5, 3, 6, -15, 20, -15, 6, 1, 7, -21, 35, -35, 21, -7, 3, 8, -28, 56, -70, 56, -28, 8, 1, 9, -36, 84, -126, 126, -84, 36, -9, 3, 10, -45, 120, -210, 252, -210, 120, -45, 10, 1

Offset: 1

Paul Curtz, Jun 25 2008

The p-th differences of a sequence a(n) are Delta^p(n) = sum_{l=0}^p (-1)^(l+p)*binomial(p,l)*a(n+l).
Setting this equal to 2*a(n) as demanded gives a recurrence with coefficients tabulated here,
a(n+p) = sum_{s=1..p} T(p,s)*a(n+p-s).

			The triangle starts in row p=0 as:
   3; (p=1, example A000244, a(n+1)=3*a(n) )
   2,  1; (p=2 example A000244 or A000129, a(n+2) = 2*a(n+1)+a(n) )
   3, -3,  3; (p=3 example A052103 or A136297, a(n+3) = 3*a(n+2)-3*a(n+1)+3*a(n) )
   4, -6,  4,   1;
   5,-10, 10,  -5,  3;
   6,-15, 20, -15,  6,   1;
   7,-21, 35, -35, 21,  -7,  3;
   8,-28, 56, -70, 56, -28,  8,  1;
   9,-36, 84,-126,126, -84, 36, -9, 3;
  10,-45,120,-210,252,-210,120,-45,10,1;

Cf. A135356.

T(p,p) = 3 if p odd, =1 if p even. T(p,s) = (-1)^(s+1)*A014410(p,s), s

Sum_{s=0..p} T(p,s) = 3.

Sum_{s=0..p} |T(p,s)| = A062510(n+1).

Edited by R. J. Mathar, Mar 02 2010

A366541 Regular triangle read by rows: T(n,k) is the number of k-facets of the bipyramid on an n-simplex base.

Original entry on oeis.org

2, 4, 4, 5, 9, 6, 6, 14, 16, 8, 7, 20, 30, 25, 10, 8, 27, 50, 55, 36, 12, 9, 35, 77, 105, 91, 49, 14, 10, 44, 112, 182, 196, 140, 64, 16, 11, 54, 156, 294, 378, 336, 204, 81, 18, 12, 65, 210, 450, 672, 714, 540, 285, 100, 20, 13, 77, 275, 660, 1122, 1386, 1254, 825, 385, 121, 22

Offset: 0

Jianing Song, Oct 12 2023

The bipyramid on an n-simplex base is the direct sum of an n-simplex and a segment. It can also be seen as two (n+1)-simplices augmented together at their base.
Note that all facets are simplices.
A deltatope is a polytope whose all cells are regular simplices (a priori not necessarily having the same size). Every polygon, being regular or not, is a 2-deltatope by definition. There are 8 convex 3-deltatopes or deltahedra (regular tetrahedron, regular octahedron, regular icosahedron, regular triangular bipyramid, regular pentagonal bipyramid and three others), 5 convex 4-deltatopes (regular 5-cell, regular 16-cell, regular 600-cell, regular tetrahedral bipyramid and regular icosahedral bipyramid) and 3 in dimension d >= 5 (regular d-simplex, regular d-orthoplex and regular bipyramid on a (d-1)-simplex base). Note that the regular orthoplex is the regular bipyramid on a hypercube base. It turns out that all cells of a deltatope are congruent (i.e., having the same size) in all nontrivial dimensions (dimension >= 3). See Dr. Richard Klitzing's answer to the Math Overflow question "4-polytopes with only one kind of regular facet" for dimension 4, and Gjergji Zaimi's answer to the question "Convex deltahedra in higher dimensions" for dimension >= 5.
More generally, a convex polytope whose all cells are regular polytopes of the same kind is either regular or a deltatope. See the article of Roswitha Blind.
The symmetry group of the bipyramid on an n-simplex base, generated by the symmetries of the n-simplex and the vertical reflexion that commute, is S_{n+1} X C_2 (with Coxeter notation [2,3^(n-1)]). See the Math Stack Exchange link.

			The triangle T(n, k) begins:
n\k   0   1    2    3     4     5     6    7    8    9  10
  0   2
  1   4   4
  2   5   9    6
  3   6  14   16    8
  4   7  20   30   25    10
  5   8  27   50   55    36    12
  6   9  35   77  105    91    49    14
  7  10  44  112  182   196   140    64   16
  8  11  54  156  294   378   336   204   81   18
  9  12  65  210  450   672   714   540  285  100   20
 10  13  77  275  660  1122  1386  1254  825  385  121  22
n = 0: the segment has 2 vertices;
n = 1: the quadrilateral has 4 vertices and 4 sides;
n = 2: the triangular bipyramid has 5 vertices, 9 edges and 6 faces;
n = 3: the tetrahedral bipyramid has 6 vertices, 14 edges, 16 faces and 8 cells.

Jianing Song, Table of n, a(n) for n = 0..5150 (Rows n = 0..100)
Math Overflow, 4-polytopes with only one kind of regular facet.
Math Overflow, Convex deltahedra in higher dimensions.
Mathematics Stack Exchange, Group symmetries of a trigonal bipyramidal molecule.
Roswitha Blind, Jürgen Tölke and Jörg M. Wills, Konvexe Polytope mit regulären Facetten im R^n (n>=4) (in German), Contributions to Geometry: Proceedings of the Geometry-Symposium held in Siegen June 28, 1978 to July 1, 1978.

A014410(n+1,k) is the number of k-facets of the n-simplex.

PARI
```
T(n,k) = if(k
				
```

T(n,k) = 2*(number of k-facets of the (n+1)-simplex) - (number of k-facets of the n-simplex) = 2*binomial(n+2,k+1) - binomial(n+1,k+1) for 0 <= k <= n-1; T(n,n) = 2*(number of n-facets of the (n+1)-simplex) - (2 cells as base) = 2*binomial(n+2,n+1) - 2 = 2*(n+1).

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A075780 Triangle T(n,k) = f(n,k,n-2), n >= 2, 1 <= k <= n-1, where f is given below.

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A075798 Triangle T(n,k) = f(n,k,n-1), n >= 0, 0 <= k <= n, where f is given below.

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A075837 Triangle T(n,k) = f(n,k,n-2), n >= 0, 0 <= k <= n, where f is given below.

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A257241 Irregular triangle read by rows: Stifel's version of the arithmetical triangle.

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A259052 Sum of Pascal triples.

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A091963 a(n) is the smallest gcd of two interior numbers on row n of Pascal's triangle ("interior" means that the 1's at the ends of the rows are excluded).

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A180986 T(n,k) = number of n X k binary matrices with rows in lexicographically nondecreasing order and columns in lexicographically nonincreasing order.

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A014719 Squares of elements in Pascal triangle (by row) that are not 1.

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