A379598 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 A110447.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 6, 0, 1, 4, 9, 16, 23, 0, 1, 5, 14, 31, 62, 104, 0, 1, 6, 20, 52, 123, 278, 531, 0, 1, 7, 27, 80, 213, 552, 1398, 2982, 0, 1, 8, 35, 116, 340, 964, 2750, 7718, 18109, 0, 1, 9, 44, 161, 513, 1561, 4784, 14976, 46083, 117545, 0

Offset: 0

Author

Seiichi Manyama, Feb 27 2025

Keywords

• nonn
• tabl

Examples

			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,   6,   16,   31,   52,   80,   116, ...
  0,  23,   62,  123,  213,  340,   513, ...
  0, 104,  278,  552,  964, 1561,  2400, ...
  0, 531, 1398, 2750, 4784, 7755, 11987, ...
		

Crossrefs

Columns k=0..1 give A000007, A110447 (A030266(n+1)).

Cf. A379599, A380178.

Programs

PARI

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

Formula

See A030266.