A384802 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 A213109.

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 22, 0, 1, 4, 15, 62, 233, 0, 1, 5, 24, 126, 696, 3716, 0, 1, 6, 35, 220, 1497, 11082, 77257, 0, 1, 7, 48, 350, 2768, 24228, 229756, 2026606, 0, 1, 8, 63, 522, 4665, 46004, 504657, 5961846, 63726497, 0, 1, 9, 80, 742, 7368, 80100, 969400, 13042326, 185814320, 2333516392, 0

Offset

0,8

Links

Table of n, a(n) for n=0..65.

Formula

Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} (-3*n+3*j+k)^(j-1) * binomial(n,j) * b(n-j,j). Then A(n,k) = b(n,-k).

Example

Square array begins:
  1,    1,     1,     1,     1,     1, ...
  0,    1,     2,     3,     4,     5, ...
  0,    3,     8,    15,    24,    35, ...
  0,   22,    62,   126,   220,   350, ...
  0,  233,   696,  1497,  2768,  4665, ...
  0, 3716, 11082, 24228, 46004, 80100, ...

Prog

(PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-3*n+3*j+k)^(j-1)*binomial(n, j)*b(n-j, j)));
a(n, k) = b(n, -k);

Crossrefs

Columns k=0..1 give A000007, A213109.

Sequence in context: A128888 A384801 A396412 * A380178 A384804 A384741

Adjacent sequences: A384799 A384800 A384801 * A384803 A384804 A384805

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 10 2025

Status

approved