A384901 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 A384896.

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 0, 0, 1, 4, 12, 6, -23, 0, 1, 5, 18, 19, -37, -51, 0, 1, 6, 25, 40, -33, -148, 27, 0, 1, 7, 33, 70, -1, -264, -186, 920, 0, 1, 8, 42, 110, 70, -360, -681, 1588, 5469, 0, 1, 9, 52, 161, 192, -384, -1446, 1437, 13469, 4836, 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,   0,    6,   19,   40,   70, ...
  0, -23,  -37,  -33,   -1,   70, ...
  0, -51, -148, -264, -360, -384, ...
		

Crossrefs

Columns k=0..1 give A000007, A384896.

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,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(-n+2*j+k-1,j-1) * b(n-j,3*j)/j. Then A(n,k) = b(n,-k).