A384804 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 A384803.

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 28, 0, 1, 4, 15, 74, 365, 0, 1, 5, 24, 144, 1008, 7456, 0, 1, 6, 35, 244, 2037, 20242, 198967, 0, 1, 7, 48, 380, 3584, 40848, 535936, 6600448, 0, 1, 8, 63, 558, 5805, 72484, 1076427, 17641290, 260641817, 0, 1, 9, 80, 784, 8880, 119200, 1909120, 35239872, 693025024, 11805179392, 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} (-4*n+4*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,   28,    74,   144,   244,    380, ...
  0,  365,  1008,  2037,  3584,   5805, ...
  0, 7456, 20242, 40848, 72484, 119200, ...

Prog

(PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-4*n+4*j+k)^(j-1)*binomial(n, j)*b(n-j, j)));
a(n, k) = b(n, -k);

Crossrefs

Columns k=0..1 give A000007, A384803.

Sequence in context: A396412 A384802 A380178 * A384741 A384742 A305401

Adjacent sequences: A384801 A384802 A384803 * A384805 A384806 A384807

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 10 2025

Status

approved