A384859 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 A384855.

1, 1, 0, 1, 1, 0, 1, 2, 7, 0, 1, 3, 16, 10, 0, 1, 4, 27, 62, -503, 0, 1, 5, 40, 162, -632, -8564, 0, 1, 6, 55, 316, -135, -20758, -103751, 0, 1, 7, 72, 530, 1264, -31572, -413900, 3479554, 0, 1, 8, 91, 810, 3865, -34316, -919647, 2636678, 327940225, 0, 1, 9, 112, 1162, 7992, -20500, -1552472, -5475222, 679001872, 8613464536, 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,     7,     16,     27,     40,     55, ...
  0,    10,     62,    162,    316,    530, ...
  0,  -503,   -632,   -135,   1264,   3865, ...
  0, -8564, -20758, -31572, -34316, -20500, ...
		

Crossrefs

Columns k=0..1 give A000007, A384855.

Cf. A384801, A384808.

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, 3*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,3*j). Then A(n,k) = b(n,-k).