A384653 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 A384649.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 9, 0, 1, 4, 9, 22, 56, 0, 1, 5, 14, 40, 134, 432, 0, 1, 6, 20, 64, 240, 1012, 3935, 0, 1, 7, 27, 95, 381, 1779, 9039, 40820, 0, 1, 8, 35, 134, 565, 2780, 15596, 92246, 471633, 0, 1, 9, 44, 182, 801, 4071, 23950, 156597, 1051558, 5980210, 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-3*j+k,j)/(4*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,    2,    5,     9,    14,    20,    27, ...
  0,    9,   22,    40,    64,    95,   134, ...
  0,   56,  134,   240,   381,   565,   801, ...
  0,  432, 1012,  1779,  2780,  4071,  5718, ...
  0, 3935, 9039, 15596, 23950, 34515, 47786, ...

Prog

(PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(4*n-3*j+k, j)/(4*n-3*j+k)*a(n-j, j)));

Crossrefs

Columns k=0..1 give A000007, A384649.

Cf. A379598, A384651, A384652, A384654.

Sequence in context: A384651 A091063 A384652 * A384654 A392379 A246935

Adjacent sequences: A384650 A384651 A384652 * A384654 A384655 A384656

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 06 2025

Status

approved