A269951 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.

1, 0, 1, 0, 2, 1, 0, 5, 5, 1, 0, 16, 23, 9, 1, 0, 65, 116, 65, 14, 1, 0, 326, 669, 470, 145, 20, 1, 0, 1957, 4429, 3634, 1415, 280, 27, 1, 0, 13700, 33375, 30681, 14084, 3535, 490, 35, 1, 0, 109601, 283072, 284066, 147532, 43939, 7756, 798, 44, 1

Offset: 0

Author

Peter Luschny, Apr 10 2016

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  0,   1;
  0,   2,   1;
  0,   5,   5,   1;
  0,  16,  23,   9,   1;
  0,  65, 116,  65,  14,  1;
  0, 326, 669, 470, 145, 20, 1;
		

Links

• Peter Luschny, Extensions of the binomial

Crossrefs

A000522 (col. 1), A073596 (col. 2), A000096 (diag. n-1), A241765 (diag. n-2).

A001339 (row sums), A137597 (unsigned matrix inverse).

Cf. A132393, A269952.

Programs

Maple

A269951 := (n,k) -> add((-1)^(n-j)*binomial(-j,-n)*abs(Stirling1(j,k)), j=0..n):
seq(seq(A269951(n,k), k=0..n), n=0..9);

Mathematica

Flatten[ Table[ Sum[(-1)^(n-j) Binomial[-j,-n] Abs[StirlingS1[j,k]], {j,0,n}], {n,0,9}, {k,0,n}]]