A384761 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A384760.

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 4, 5, 0, 1, -4, 9, 4, -35, 0, 1, -5, 16, -9, -104, -281, 0, 1, -6, 25, -40, -171, -112, 5671, 0, 1, -7, 36, -95, -176, 717, 14164, 42671, 0, 1, -8, 49, -180, -35, 2176, 20619, -18104, -2179127, 0

Offset: 0

Author

Seiichi Manyama, Jun 09 2025

Keywords

• sign
• tabl

Examples

			Square array begins:
  1,    1,    1,    1,    1,    1, ...
  0,   -1,   -2,   -3,   -4,   -5, ...
  0,    1,    4,    9,   16,   25, ...
  0,    5,    4,   -9,  -40,  -95, ...
  0,  -35, -104, -171, -176,  -35, ...
  0, -281, -112,  717, 2176, 3875, ...
		

Crossrefs

Columns k=0..1 give A000007, A384760.

Cf. A379168.

Programs

PARI

a(n, k) = if(k==0, 0^n, (-1)^n*k*sum(j=0, n, (n+k)^(j-1)*binomial(n, j)*a(n-j, j)));

Formula

A(n,0) = 0^n; A(n,k) = (-1)^n * k * Sum_{j=0..n} (n+k)^(j-1) * binomial(n,j) * A(n-j,j).