A384860 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 A384856.

1, 1, 0, 1, 1, 0, 1, 2, 7, 0, 1, 3, 16, 28, 0, 1, 4, 27, 98, -107, 0, 1, 5, 40, 216, 304, -11744, 0, 1, 6, 55, 388, 1485, -20638, -519101, 0, 1, 7, 72, 620, 3712, -20592, -1185920, -12366080, 0, 1, 8, 91, 918, 7285, -3836, -1908657, -35662030, -101065751, 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,     28,     98,    216,   388,   620, ...
  0,   -107,    304,   1485,  3712,  7285, ...
  0, -11744, -20638, -20592, -3836, 39200, ...
		

Crossrefs

Columns k=0..1 give A000007, A384856.

Cf. A058127, A384811.

Programs

PARI

b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-2*n+2*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} (-2*n+2*j+k)^(j-1) * binomial(n,j) * b(n-j,3*j). Then A(n,k) = b(n,-k).