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
KEYWORD
nonn,tabl
AUTHOR
Thomas Wieder, Dec 29 2006
STATUS
approved