A269954 Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.

1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 2, 5, 3, 1, 0, 9, 20, 17, 6, 1, 0, 44, 109, 100, 45, 10, 1, 0, 265, 689, 694, 355, 100, 15, 1, 0, 1854, 5053, 5453, 3094, 1015, 196, 21, 1, 0, 14833, 42048, 48082, 29596, 10899, 2492, 350, 28, 1

Offset: 0

Author

Peter Luschny, Apr 12 2016

Keywords

• nonn
• tabl

Examples

			Triangle starts:
  1;
  0,   1;
  0,   0,   1;
  0,   1,   1,   1;
  0,   2,   5,   3,   1;
  0,   9,  20,  17,   6,   1;
  0,  44, 109, 100,  45,  10,  1;
  0, 265, 689, 694, 355, 100, 15, 1;
		

Links

• Peter Luschny, Extensions of the binomial

Crossrefs

A000255 (row sums), A000217 (diag. n,n-1), A133252 (diag. n,n-2).

Columns k=0..4 give A000007, A000166(n-1), A300490(n-1), A381067(n-1), A381068(n-1).

Cf. A269951, A269953.

Programs

Maple

A269954 := (n, k) -> add(binomial(-j, -n)*abs(Stirling1(j, k)), j=0..n):
seq(seq(A269954(n, k), k=0..n), n=0..9);

Mathematica

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

PARI

T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(n-1, n-j)*abs(stirling(j, k)));
for(n=0, 9, for(k=0, n, print1(T(n, k), ", "))); \\ Seiichi Manyama, Feb 13 2025