A385061 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 A385058.

1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 12, 31, 0, 1, 4, 21, 92, 333, 0, 1, 5, 32, 189, 1064, 3841, 0, 1, 6, 45, 328, 2373, 14112, 57463, 0, 1, 7, 60, 515, 4464, 34923, 230188, 836109, 0, 1, 8, 77, 756, 7565, 71584, 615195, 4005920, 11138921, 0, 1, 9, 96, 1057, 11928, 130725, 1351384, 11934219, 72843408, 14908465, 0

Offset

0,8

Comments

A(10,1) = -10091931669.

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} (-n+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,    5,    12,    21,    32,     45, ...
  0,   31,    92,   189,   328,    515, ...
  0,  333,  1064,  2373,  4464,   7565, ...
  0, 3841, 14112, 34923, 71584, 130725, ...

Prog

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

Crossrefs

Columns k=0..1 give A000007, A385058.

Sequence in context: A381648 A384808 A384811 * A384813 A384985 A396994

Adjacent sequences: A385058 A385059 A385060 * A385062 A385063 A385064

Keyword

sign,tabl

Author

Seiichi Manyama, Jun 16 2025

Status

approved