A384944 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 A384941.

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, -2, 0, 1, 4, 15, 4, -64, 0, 1, 5, 22, 19, -116, -95, 0, 1, 6, 30, 44, -144, -334, 780, 0, 1, 7, 39, 80, -135, -675, 862, 5230, 0, 1, 8, 49, 128, -75, -1060, 70, 11516, 19228, 0, 1, 9, 60, 189, 51, -1414, -1684, 16953, 59632, -90488, 0

Offset: 0

Author

Seiichi Manyama, Jun 13 2025

Keywords

• sign
• tabl

Examples

			Square array begins:
  1,   1,    1,    1,     1,     1,     1, ...
  0,   1,    2,    3,     4,     5,     6, ...
  0,   4,    9,   15,    22,    30,    39, ...
  0,  -2,    4,   19,    44,    80,   128, ...
  0, -64, -116, -144,  -135,   -75,    51, ...
  0, -95, -334, -675, -1060, -1414, -1644, ...
  0, 780,  862,   70, -1684, -4380, -7869, ...
		

Crossrefs

Columns k=0..1 give A000007, A384941.

Cf. A384899, A384901.

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, 4*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,4*j)/j. Then A(n,k) = b(n,-k).