A126350 -id:A126350 - cp's OEIS frontend

A054654 Triangle of Stirling numbers of 1st kind, S(n, n-k), n >= 0, 0 <= k <= n.

1, 1, 0, 1, -1, 0, 1, -3, 2, 0, 1, -6, 11, -6, 0, 1, -10, 35, -50, 24, 0, 1, -15, 85, -225, 274, -120, 0, 1, -21, 175, -735, 1624, -1764, 720, 0, 1, -28, 322, -1960, 6769, -13132, 13068, -5040, 0

Offset: 0

N. J. A. Sloane, Apr 18 2000

Triangle is the matrix product of the binomial coefficients with the Stirling numbers of the first kind.
Triangle T(n,k) giving coefficients in expansion of n!*C(x,n) in powers of x. E.g., 3!*C(x,3) = x^3-3*x^2+2*x.
The matrix product of binomial coefficients with the Stirling numbers of the first kind results in the Stirling numbers of the first kind again, but the triangle is shifted by (1,1).
Essentially [1,0,1,0,1,0,1,0,...] DELTA [0,-1,-1,-2,-2,-3,-3,-4,-4,...] where DELTA is the operator defined in A084938; mirror image of the Stirling-1 triangle A048994. - Philippe Deléham, Dec 30 2006
From Doudou Kisabaka, Dec 18 2009: (Start)
The sum of the entries on each row of the triangle, starting on the 3rd row, equals 0. E.g., 1+(-3)+2+0 = 0.
The entries on the triangle can be computed as follows. T(n,r) = T(n-1,r) - (n-1)*T(n-1,r-1). T(n,r) = 0 when r equals 0 or r > n. T(n,r) = 1 if n==1. (End)

			Matrix begins:
  1, 0,  0,  0,  0,   0,    0,     0,      0, ...
  0, 1, -1,  2, -6,  24, -120,   720,  -5040, ...
  0, 0,  1, -3, 11, -50,  274, -1764,  13068, ...
  0, 0,  0,  1, -6,  35, -225,  1624, -13132, ...
  0, 0,  0,  0,  1, -10,   85,  -735,   6769, ...
  0, 0,  0,  0,  0,   1,  -15,   175,  -1960, ...
  0, 0,  0,  0,  0,   0,    1,   -21,    322, ...
  0, 0,  0,  0,  0,   0,    0,     1,    -28, ...
  0, 0,  0,  0,  0,   0,    0,     0,      1, ...
  ...
Triangle begins:
  1;
  1,   0;
  1,  -1,   0;
  1,  -3,   2,    0;
  1,  -6,  11,   -6,    0;
  1, -10,  35,  -50,   24,     0;
  1, -15,  85, -225,  274,  -120,   0;
  1, -21, 175, -735, 1624, -1764, 720, 0;
  ...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 18, table 18:6:1 at page 152.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
Eric Weisstein's World of Mathematics, Rising Factorial.
Eric Weisstein's World of Mathematics, FallingFactorial.

Essentially Stirling numbers of first kind, multiplied by factorials - see A008276.
The Stirling2 counterpart is A106800.
Cf. A054655, A039810, A039814, A126350, A126351, A126353, A340556.

a054654 n k = a054654_tabl !! n !! k
a054654_row n = a054654_tabl !! n
a054654_tabl = map reverse a048994_tabl
-- Reinhard Zumkeller, Mar 18 2014

a054654_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (-x,n)),x,n-k),k=0..n) end: # Peter Luschny, Nov 28 2010
seq(seq(Stirling1(n, n-k), k=0..n), n=0..8); # Peter Luschny, Feb 21 2021

row[n_] := Reverse[ CoefficientList[ (-1)^n*Pochhammer[-x, n], x] ]; Flatten[ Table[ row[n], {n, 0, 8}]] (* Jean-François Alcover, Feb 16 2012, after Maple *)
Table[StirlingS1[n,n-k],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Jun 17 2023 *)

PARI
```
T(n,k)=polcoeff(n!*binomial(x,n), n-k)
```

n!*binomial(x, n) = Sum_{k=0..n} T(n, k)*x^(n-k).
(In Maple notation:) Matrix product A*B of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling1(j,i) with i from 1 to d, j from 1 to d, d=9.
T(n, k) = (-1)^k*Sum_{j=0..k} E2(k, j)*binomial(n+j-1, 2*k), where E2(k, j) are the second-order Eulerian numbers A340556. - Peter Luschny, Feb 21 2021

Additional comments from Thomas Wieder, Dec 29 2006

Edited by N. J. A. Sloane at the suggestion of Eric W. Weisstein, Jan 20 2008

A126351 Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.

Original entry on oeis.org

1, 1, 2, 1, 5, 4, 1, 9, 19, 8, 1, 14, 55, 65, 16, 1, 20, 125, 285, 211, 32, 1, 27, 245, 910, 1351, 665, 64, 1, 35, 434, 2380, 5901, 6069, 2059, 128, 1, 44, 714, 5418, 20181, 35574, 26335, 6305, 256, 1, 54, 1110, 11130, 58107, 156660, 204205, 111645, 19171, 512

Offset: 1

Thomas Wieder, Dec 29 2006

Many well-known integer sequences arise from such a matrix product of combinatorial coefficients. In the present case we have as the first row A000079 = the powers of two = 2^n. As the second row we have A001047 = 3^n - 2^n. As the column sums we have 1,3,10,37,151,674,3263,17007,94828 we have A005493 = number of partitions of [n+1] with a distinguished block.

			Matrix begins:
1, 2, 4,  8, 16,  32,   64,  128,   256, ... A000079
0, 1, 5, 19, 65, 211,  665, 2059,  6305, ... A001047
0, 0, 1,  9, 55, 285, 1351, 6069, 26335, ... A016269
0, 0, 0,  1, 14, 125,  910, 5901, 35574, ... A025211
0, 0, 0,  0,  1,  20,  245, 2380, 20181, ...
0, 0, 0,  0,  0,   1,   27,  434,  5418, ...
0, 0, 0,  0,  0,   0,    1,   35,   714, ...
0, 0, 0,  0,  0,   0,    0,    1,    44, ...
0, 0, 0,  0,  0,   0,    0,    0,     1, ...
Triangle begins:
1;
1,  2;
1,  5,  4;
1,  9, 19,  8;
1, 14, 55, 65, 16;

Alois P. Heinz, Rows n = 1..100, flattened

Cf. A039810, A039814, A126350, A054654, A126353.

T:= (n, k)-> add(binomial(n-1, i-1) *Stirling2(i, n+1-k), i=1..n):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 29 2011

T[n_, k_] := Sum[Binomial[n-1, i-1]*StirlingS2[i, n+1-k], {i, 1, n}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

(In Maple notation:) Matrix product B.A of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling2(j,i) with i from 1 to d, j from 1 to d, d=9.

T(n,k) = Sum_{i=1..n} C(n-1,i-1) * Stirling2(i, n+1-k). - Alois P. Heinz, Sep 29 2011

A126353 Triangle read by rows: matrix product of the Stirling numbers of the first kind with the binomial coefficients.

Original entry on oeis.org

1, 1, 0, 1, -1, 1, 1, -3, 5, -2, 1, -6, 17, -20, 9, 1, -10, 45, -100, 109, -44, 1, -15, 100, -355, 694, -689, 265, 1, -21, 196, -1015, 3094, -5453, 5053, -1854, 1, -28, 350, -2492, 10899, -29596, 48082, -42048, 14833

Offset: 1

Thomas Wieder, Dec 29 2006

Many well-known integer sequences arise from such a matrix product of combinatorial coefficients. In the present case we have as the first row A000166 = subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.

			Matrix begins:
1 0 1 -2 9 -44 265 -1854 14833
0 1 -1 5 -20 109 -689 5053 -42048
0 0 1 -3 17 -100 694 -5453 48082
0 0 0 1 -6 45 -355 3094 -29596
0 0 0 0 1 -10 100 -1015 10899
0 0 0 0 0 1 -15 196 -2492
0 0 0 0 0 0 1 -21 350
0 0 0 0 0 0 0 1 -28
0 0 0 0 0 0 0 0 1

Signed version of A094791 [from Olivier Gérard, Jul 31 2011]

Cf. A039810, A039814, A126350, A126351, A054654.

(In Maple notation:) Matrix product B.A of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling1(j,i) with i from 1 to d, j from 1 to d, d=9.

A137597 Triangle read by rows: A008277 * A007318.

Original entry on oeis.org

1, 2, 1, 5, 5, 1, 15, 22, 9, 1, 52, 99, 61, 14, 1, 203, 471, 385, 135, 20, 1, 877, 2386, 2416, 1140, 260, 27, 1, 4140, 12867, 15470, 9156, 2835, 455, 35, 1, 21147, 73681, 102215, 72590, 28441, 6230, 742, 44, 1

Offset: 1

Gary W. Adamson, Jan 29 2008

Row sums = A035009 starting (1, 3, 11, 47, 227, ...).

			First few rows of the triangle:
    1;
    2,   1;
    5,   5,   1;
   15,  22,   9,   1;
   52,  99,  61,  14,  1;
  203, 471, 385, 135, 20, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See p. 16.

Cf. A035009, A008277.

Cf. A126350.

T:= (n, k)-> add(Stirling2(n, j)*binomial(j-1, k-1), j=k..n):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 03 2019

Table[Sum[StirlingS2[n, j]*Binomial[j - 1, k - 1], {j, k, n}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Aug 31 2023 *)

A008277 * A007318 as infinite lower triangular matrices.

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A054654 Triangle of Stirling numbers of 1st kind, S(n, n-k), n >= 0, 0 <= k <= n.

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A126351 Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.

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A126353 Triangle read by rows: matrix product of the Stirling numbers of the first kind with the binomial coefficients.

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Formula

A137597 Triangle read by rows: A008277 * A007318.

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