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

%I A126350 #19 Mar 17 2020 13:50:02
%S A126350 1,1,2,1,5,5,1,9,22,15,1,14,61,99,52,1,20,135,385,471,203,1,27,260,
%T A126350 1140,2416,2386,877,1,35,455,2835,9156,15470,12867,4140,1,44,742,6230,
%U A126350 28441,72590,102215,73681,21147
%N A126350 Triangle read by rows: matrix product of the binomial coefficients with the Stirling numbers of the second kind.
%C A126350 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, ...].
%F A126350 (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.
%e A126350 Matrix begins:
%e A126350 1 2 5 15 52 203  877  4140  21147
%e A126350 0 1 5 22 99 471 2386 12867  73681
%e A126350 0 0 1  9 61 385 2416 15470 102215
%e A126350 0 0 0  1 14 135 1140  9156  72590
%e A126350 0 0 0  0 1   20  260  2835  28441
%e A126350 0 0 0  0 0    1   27   455   6230
%e A126350 0 0 0  0 0    0    1    35    742
%e A126350 0 0 0  0 0    0    0     1     44
%e A126350 0 0 0  0 0    0    0     0      1
%p A126350 T:= (n, k)-> add(Stirling2(n, j)*binomial(j-1, n-k), j=n-k+1..n):
%p A126350 seq(seq(T(n, k), k=1..n), n=1..10);  # _Alois P. Heinz_, Sep 03 2019
%t A126350 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}]];
%t A126350 dim = 9;
%t A126350 Table[T[dim][[n]][[1 ;; n]] // Reverse, {n, 1, dim}] (* _Jean-François Alcover_, Jun 27 2019, from Sage *)
%o A126350 (Sage)
%o A126350 def A126350_triangle(dim): # rows in reversed order
%o A126350     M = matrix(ZZ,dim,dim)
%o A126350     for n in (0..dim-1): M[n,n] = 1
%o A126350     for n in (1..dim-1):
%o A126350         for k in (0..n-1):
%o A126350             M[n,k] = M[n-1,k-1]+(k+2)*M[n-1,k]+(k+1)*M[n-1,k+1]
%o A126350     return M
%o A126350 A126350_triangle(9) # _Peter Luschny_, Sep 19 2012
%Y A126350 Cf. A039810, A039814, A126351, A054654, A126353.
%Y A126350 Cf. A137597.
%K A126350 nonn,tabl
%O A126350 1,3
%A A126350 _Thomas Wieder_, Dec 29 2006