A384808 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 A384617.

1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 12, 13, 0, 1, 4, 21, 56, -63, 0, 1, 5, 32, 135, 128, -2279, 0, 1, 6, 45, 256, 753, -3888, -51167, 0, 1, 7, 60, 425, 2016, -1797, -135752, -423387, 0, 1, 8, 77, 648, 4145, 8224, -224775, -2099032, 13717889, 0, 1, 9, 96, 931, 7392, 31725, -256016, -5236809, 3294432, 885044593, 0

Offset: 0

Author

Seiichi Manyama, Jun 10 2025

Keywords

• sign
• tabl

Examples

			Square array begins:
  1,     1,     1,     1,    1,     1, ...
  0,     1,     2,     3,    4,     5, ...
  0,     5,    12,    21,   32,    45, ...
  0,    13,    56,   135,  256,   425, ...
  0,   -63,   128,   753, 2016,  4145, ...
  0, -2279, -3888, -1797, 8224, 31725, ...
		

Crossrefs

Columns k=0..1 give A000007, A384617.

Programs

PARI

b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-n+j+k)^(j-1)*binomial(n, j)*b(n-j, 2*j)));
a(n, k) = b(n, -k);

Formula

Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} (-n+j+k)^(j-1) * binomial(n,j) * b(n-j,2*j). Then A(n,k) = b(n,-k).