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Triangle read by rows: matrix product of the binomial coefficients with the Stirling numbers of the second kind.
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%I #19 Mar 17 2020 13:50:02

%S 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 1140,2416,2386,877,1,35,455,2835,9156,15470,12867,4140,1,44,742,6230,

%U 28441,72590,102215,73681,21147

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

%C 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 (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 Matrix begins:

%e 1 2 5 15 52 203 877 4140 21147

%e 0 1 5 22 99 471 2386 12867 73681

%e 0 0 1 9 61 385 2416 15470 102215

%e 0 0 0 1 14 135 1140 9156 72590

%e 0 0 0 0 1 20 260 2835 28441

%e 0 0 0 0 0 1 27 455 6230

%e 0 0 0 0 0 0 1 35 742

%e 0 0 0 0 0 0 0 1 44

%e 0 0 0 0 0 0 0 0 1

%p T:= (n, k)-> add(Stirling2(n, j)*binomial(j-1, n-k), j=n-k+1..n):

%p seq(seq(T(n, k), k=1..n), n=1..10); # _Alois P. Heinz_, Sep 03 2019

%t 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 dim = 9;

%t Table[T[dim][[n]][[1 ;; n]] // Reverse, {n, 1, dim}] (* _Jean-François Alcover_, Jun 27 2019, from Sage *)

%o (Sage)

%o def A126350_triangle(dim): # rows in reversed order

%o M = matrix(ZZ,dim,dim)

%o for n in (0..dim-1): M[n,n] = 1

%o for n in (1..dim-1):

%o for k in (0..n-1):

%o M[n,k] = M[n-1,k-1]+(k+2)*M[n-1,k]+(k+1)*M[n-1,k+1]

%o return M

%o A126350_triangle(9) # _Peter Luschny_, Sep 19 2012

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

%Y Cf. A137597.

%K nonn,tabl

%O 1,3

%A _Thomas Wieder_, Dec 29 2006