A384758 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 A384757.

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, 0, 14, 0, 1, -4, 3, 34, 9, 0, 1, -5, 8, 54, -88, -1516, 0, 1, -6, 15, 68, -327, -3402, 4345, 0, 1, -7, 24, 70, -720, -4908, 30532, 507870, 0, 1, -8, 35, 54, -1255, -5044, 84321, 1027402, -4984063, 0

Offset

0,8

Links

Table of n, a(n) for n=0..54.

Formula

A(n,0) = 0^n; A(n,k) = (-1)^n * k * Sum_{j=0..n} (n-j+k)^(j-1) * binomial(n,j) * A(n-j,j).

Example

Square array begins:
  1,     1,     1,     1,     1,     1, ...
  0,    -1,    -2,    -3,    -4,    -5, ...
  0,    -1,     0,     3,     8,    15, ...
  0,    14,    34,    54,    68,    70, ...
  0,     9,   -88,  -327,  -720, -1255, ...
  0, -1516, -3402, -4908, -5044, -2700, ...

Prog

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

Crossrefs

Columns k=0..1 give A000007, A384757.

Cf. A384761.

Sequence in context: A131185 A307819 A396860 * A286354 A296067 A306713

Adjacent sequences: A384755 A384756 A384757 * A384759 A384760 A384761

Keyword

sign,tabl

Author

Seiichi Manyama, Jun 09 2025

Status

approved