A384581 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 A143501.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 4, 0, 1, 4, 6, 10, 16, 0, 1, 5, 10, 19, 41, 92, 0, 1, 6, 15, 32, 78, 224, 616, 0, 1, 7, 21, 50, 131, 411, 1464, 4729, 0, 1, 8, 28, 74, 205, 672, 2617, 11002, 40776, 0, 1, 9, 36, 105, 306, 1031, 4170, 19251, 93234, 388057, 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(3*n-3*j+k,j)/(3*n-3*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,   4,   10,   19,   32,   50,   74, ...
  0,  16,   41,   78,  131,  205,  306, ...
  0,  92,  224,  411,  672, 1031, 1518, ...
  0, 616, 1464, 2617, 4170, 6245, 8997, ...

Prog

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

Crossrefs

Columns k=0..1 give A000007, A143501.

Cf. A381566, A384580, A384582, A384583.

Sequence in context: A392378 A261835 A384866 * A384582 A384583 A353436

Adjacent sequences: A384578 A384579 A384580 * A384582 A384583 A384584

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 04 2025

Status

approved