A384899 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 A384894.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 1, 0, 1, 4, 9, 6, -4, 0, 1, 5, 14, 16, -2, -14, 0, 1, 6, 20, 32, 12, -32, -30, 0, 1, 7, 27, 55, 45, -39, -103, 12, 0, 1, 8, 35, 86, 105, -12, -211, -100, 330, 0, 1, 9, 44, 126, 201, 81, -318, -411, 552, 1139, 0, 1, 10, 54, 176, 343, 282, -350, -956, 342, 3038, 2226, 0

Offset: 0

Author

Seiichi Manyama, Jun 12 2025

Keywords

• sign
• tabl

Examples

			Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   1,   2,   3,   4,   5, ...
  0,   2,   5,   9,  14,  20, ...
  0,   1,   6,  16,  32,  55, ...
  0,  -4,  -2,  12,  45, 105, ...
  0, -14, -32, -39, -12,  81, ...
		

Crossrefs

Columns k=0..1 give A000007, A384894.

Programs

PARI

b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-n+2*j+k-1,j-1)*b(n-j,2*j)/j));
a(n, k) = b(n, -k);

Formula

Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-n+2*j+k-1,j-1) * b(n-j,2*j)/j. Then A(n,k) = b(n,-k).