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 A126351 Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients. 5
 1, 1, 2, 1, 5, 4, 1, 9, 19, 8, 1, 14, 55, 65, 16, 1, 20, 125, 285, 211, 32, 1, 27, 245, 910, 1351, 665, 64, 1, 35, 434, 2380, 5901, 6069, 2059, 128, 1, 44, 714, 5418, 20181, 35574, 26335, 6305, 256, 1, 54, 1110, 11130, 58107, 156660, 204205, 111645, 19171, 512 (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 A000079 = the powers of two = 2^n. As the second row we have A001047 = 3^n - 2^n. As the column sums we have 1,3,10,37,151,674,3263,17007,94828 we have A005493 = number of partitions of [n+1] with a distinguished block. LINKS Alois P. Heinz, Rows n = 1..100, flattened FORMULA (In Maple notation:) Matrix product B.A 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. T(n,k) = Sum_{i=1..n} C(n-1,i-1) * Stirling2(i, n+1-k). - Alois P. Heinz, Sep 29 2011 EXAMPLE Matrix begins: 1, 2, 4,  8, 16,  32,   64,  128,   256, ... A000079 0, 1, 5, 19, 65, 211,  665, 2059,  6305, ... A001047 0, 0, 1,  9, 55, 285, 1351, 6069, 26335, ... A016269 0, 0, 0,  1, 14, 125,  910, 5901, 35574, ... A025211 0, 0, 0,  0,  1,  20,  245, 2380, 20181, ... 0, 0, 0,  0,  0,   1,   27,  434,  5418, ... 0, 0, 0,  0,  0,   0,    1,   35,   714, ... 0, 0, 0,  0,  0,   0,    0,    1,    44, ... 0, 0, 0,  0,  0,   0,    0,    0,     1, ... Triangle begins: 1; 1,  2; 1,  5,  4; 1,  9, 19,  8; 1, 14, 55, 65, 16; MAPLE T:= (n, k)-> add(binomial(n-1, i-1) *Stirling2(i, n+1-k), i=1..n): seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 29 2011 MATHEMATICA T[n_, k_] := Sum[Binomial[n-1, i-1]*StirlingS2[i, n+1-k], {i, 1, n}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *) CROSSREFS Cf. A039810, A039814, A126350, A054654, A126353. Sequence in context: A056242 A128718 A112358 * A157011 A246173 A092821 Adjacent sequences:  A126348 A126349 A126350 * A126352 A126353 A126354 KEYWORD nonn,tabl AUTHOR Thomas Wieder, Dec 29 2006 STATUS approved

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Last modified April 14 12:11 EDT 2021. Contains 342949 sequences. (Running on oeis4.)