A271700 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j-1,-n-1)*S1(k,j), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.

1, 1, 1, 1, 2, 3, 1, 3, 6, 16, 1, 4, 10, 30, 115, 1, 5, 15, 50, 205, 1021, 1, 6, 21, 77, 336, 1750, 10696, 1, 7, 28, 112, 518, 2814, 17766, 128472, 1, 8, 36, 156, 762, 4308, 28050, 207942, 1734447, 1, 9, 45, 210, 1080, 6342, 42528, 322860, 2746815, 25937683

Offset: 0

Author

Peter Luschny, Apr 14 2016

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[1]
[1, 1]
[1, 2, 3]
[1, 3, 6,  16]
[1, 4, 10, 30,  115]
[1, 5, 15, 50,  205, 1021]
[1, 6, 21, 77,  336, 1750, 10696]
[1, 7, 28, 112, 518, 2814, 17766, 128472]
		

Crossrefs

A000027 (col. 1), A000217, A161680 (col. 2), A005581 (col. 3), A211210 (diag. n,n), A211211 (diag. n,n-1).

Programs

Maple

T := (n,k) -> add(abs(Stirling1(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] Abs[StirlingS1[k,j]],{j,0,n}], {n,0,9},{k,0,n}]]