A126350 Triangle read by rows: matrix product of the binomial coefficients with the Stirling numbers of the second kind.

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

Offset

1,3

Comments

Many well-known integer sequences arise from such a matrix product of combinatorial coefficients. In the present case we have as the first row (not surprisingly) A000110 = Bell or exponential numbers: ways of placing n labeled balls into n indistinguishable boxes . As second row we have A033452 = "STIRLING" transform of squares A000290. As the column sums we have 1, 3, 11, 47, 227, 1215, 7107, 44959, 305091 which is A035009 = STIRLING transform of [1,1,2,4,8,16,32, ...].

Links

Table of n, a(n) for n=1..45.

Formula

(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]:=stirling2(j,i) with i from 1 to d, j from 1 to d, d=9.

Example

Matrix begins:
1 2 5 15 52 203  877  4140  21147
0 1 5 22 99 471 2386 12867  73681
0 0 1  9 61 385 2416 15470 102215
0 0 0  1 14 135 1140  9156  72590
0 0 0  0 1   20  260  2835  28441
0 0 0  0 0    1   27   455   6230
0 0 0  0 0    0    1    35    742
0 0 0  0 0    0    0     1     44
0 0 0  0 0    0    0     0      1

Maple

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

Mathematica

T[dim_] := T[dim] = Module[{M}, M[n_, n_] = 1; M[_, _] = 0; Do[M[n, k] = M[n-1, k-1] + (k+2) M[n-1, k] + (k+1) M[n-1, k+1], {n, 0, dim-1}, {k, 0, n-1}]; Array[M, {dim, dim}, {0, 0}]];
dim = 9;
Table[T[dim][[n]][[1 ;; n]] // Reverse, {n, 1, dim}] (* Jean-François Alcover, Jun 27 2019, from Sage *)

Prog

(Sage)
def A126350_triangle(dim): # rows in reversed order
    M = matrix(ZZ, dim, dim)
    for n in (0..dim-1): M[n, n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n, k] = M[n-1, k-1]+(k+2)*M[n-1, k]+(k+1)*M[n-1, k+1]
    return M
A126350_triangle(9) # Peter Luschny, Sep 19 2012

Crossrefs

Cf. A039810, A039814, A126351, A054654, A126353.

Cf. A137597.

Sequence in context: A209830 A209695 A033282 * A204111 A079502 A209164

Adjacent sequences: A126347 A126348 A126349 * A126351 A126352 A126353

Keyword

nonn,tabl

Author

Thomas Wieder, Dec 29 2006

Status

approved