A381573 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 A381574.

1, 1, 0, 1, 3, 0, 1, 6, 15, 0, 1, 9, 39, 118, 0, 1, 12, 72, 326, 1206, 0, 1, 15, 114, 651, 3345, 14712, 0, 1, 18, 165, 1120, 6822, 40200, 204385, 0, 1, 21, 225, 1760, 12123, 81675, 547146, 3143826, 0, 1, 24, 294, 2598, 19815, 145968, 1096080, 8239938, 52580328, 0

Offset

0,5

Links

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

Formula

A(n,0) = 0^n; A(n,k) = 3*k * Sum_{j=0..n} binomial(3*n-2*j+3*k,j)/(3*n-2*j+3*k) * A(n-j,j).

Example

Square array begins:
  1,     1,     1,     1,      1,      1, ...
  0,     3,     6,     9,     12,     15, ...
  0,    15,    39,    72,    114,    165, ...
  0,   118,   326,   651,   1120,   1760, ...
  0,  1206,  3345,  6822,  12123,  19815, ...
  0, 14712, 40200, 81675, 145968, 241773, ...

Prog

(PARI) a(n, k) = if(k==0, 0^n, 3*k*sum(j=0, n, binomial(3*n-2*j+3*k, j)/(3*n-2*j+3*k)*a(n-j, j)));

Crossrefs

Columns k=0..1 give A000007, A381574.

Cf. A379598, A381571.

Sequence in context: A381569 A341856 A339350 * A396431 A244118 A273155

Adjacent sequences: A381570 A381571 A381572 * A381574 A381575 A381576

Keyword

nonn,tabl

Author

Seiichi Manyama, Feb 28 2025

Status

approved