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

Author

Thomas Wieder, Dec 29 2006

Keywords

• nonn
• tabl

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, ...].

Examples

			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
		

Crossrefs

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

Cf. A137597.

Programs

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 *)

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

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.