A384582 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384574.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 5, 0, 1, 4, 6, 12, 23, 0, 1, 5, 10, 22, 57, 155, 0, 1, 6, 15, 36, 105, 366, 1236, 0, 1, 7, 21, 55, 171, 651, 2853, 11286, 0, 1, 8, 28, 80, 260, 1032, 4951, 25584, 116333, 0, 1, 9, 36, 112, 378, 1536, 7656, 43587, 259789, 1329433, 0

Offset

0,8

Links

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

Formula

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(4*n-4*j+k,j)/(4*n-4*j+k) * A(n-j,j).

Example

Square array begins:
  1,    1,    1,    1,    1,     1,     1, ...
  0,    1,    2,    3,    4,     5,     6, ...
  0,    1,    3,    6,   10,    15,    21, ...
  0,    5,   12,   22,   36,    55,    80, ...
  0,   23,   57,  105,  171,   260,   378, ...
  0,  155,  366,  651, 1032,  1536,  2196, ...
  0, 1236, 2853, 4951, 7656, 11125, 15552, ...

Prog

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

Crossrefs

Columns k=0..1 give A000007, A384574.

Cf. A381566, A384580, A384581, A384583.

Sequence in context: A261835 A384866 A384581 * A384583 A353436 A286932

Adjacent sequences: A384579 A384580 A384581 * A384583 A384584 A384585

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 04 2025

Status

approved