A271702 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j-1,-n-1)*S2(k,j), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

1, 1, 1, 1, 2, 3, 1, 3, 6, 13, 1, 4, 10, 26, 71, 1, 5, 15, 45, 140, 456, 1, 6, 21, 71, 246, 887, 3337, 1, 7, 28, 105, 399, 1568, 6405, 27203, 1, 8, 36, 148, 610, 2584, 11334, 51564, 243203, 1, 9, 45, 201, 891, 4035, 18849, 91101, 455712, 2357356

Offset: 0

Author

Peter Luschny, Apr 14 2016

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[1]
[1, 1]
[1, 2, 3]
[1, 3, 6, 13]
[1, 4, 10, 26, 71]
[1, 5, 15, 45, 140, 456]
[1, 6, 21, 71, 246, 887, 3337]
[1, 7, 28, 105, 399, 1568, 6405, 27203]
		

Crossrefs

A000012 (col. 0), A000027 (col. 1), A000217 (col. 2), A008778 (col. 3), A122455 (diag. n,n), A134094 (diag. n,n-1).

Cf. A048993.

Programs

Maple

T := (n,k) -> add(Stirling2(k,j)*binomial(-j-1,-n-1)*(-1)^(n-j),j=0..n):
seq(seq(T(n,k), k=0..n), n=0..9);

Mathematica

Flatten[Table[Sum[(-1)^(n-j) Binomial[-j-1,-n-1] StirlingS2[k,j], {j,0,n}], {n,0,9}, {k,0,n}]]

Formula

T(n,k) = Sum_{j=0..k} C(n,j) * S2(k,j). - Alois P. Heinz, Sep 03 2019