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 A126350 Triangle read by rows: matrix product of the binomial coefficients with the Stirling numbers of the second kind. 4
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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

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Last modified April 17 11:26 EDT 2021. Contains 343064 sequences. (Running on oeis4.)