A384902 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 A384897.

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 3, 0, 1, 4, 12, 12, -8, 0, 1, 5, 18, 28, -1, -81, 0, 1, 6, 25, 52, 30, -160, -462, 0, 1, 7, 33, 85, 95, -201, -1125, -1140, 0, 1, 8, 42, 128, 205, -156, -1932, -3738, 1662, 0, 1, 9, 52, 182, 372, 36, -2760, -8073, -2150, 42210, 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,   3,    7,   12,   18,  25, ...
  0,   3,   12,   28,   52,  85, ...
  0,  -8,   -1,   30,   95, 205, ...
  0, -81, -160, -201, -156,  36, ...
		

Crossrefs

Columns k=0..1 give A000007, A384897.

Programs

PARI

b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-2*n+3*j+k-1,j-1)*b(n-j,3*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(-2*n+3*j+k-1,j-1) * b(n-j,3*j)/j. Then A(n,k) = b(n,-k).