A384946 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 A384943.

1, 1, 0, 1, 1, 0, 1, 2, 6, 0, 1, 3, 13, -9, 0, 1, 4, 21, -6, -244, 0, 1, 5, 30, 10, -470, -39, 0, 1, 6, 40, 40, -660, -674, 11262, 0, 1, 7, 51, 85, -795, -1824, 19599, 36971, 0, 1, 8, 63, 146, -855, -3384, 24171, 100390, -268890, 0, 1, 9, 76, 224, -819, -5224, 24318, 180627, -268456, -3724293, 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,     6,    13,    21,    30,    40,    51, ...
  0,    -9,    -6,    10,    40,    85,   146, ...
  0,  -244,  -470,  -660,  -795,  -855,  -819, ...
  0,   -39,  -674, -1824, -3384, -5224, -7188, ...
  0, 11262, 19599, 24171, 24318, 19590,  9778, ...
		

Crossrefs

Columns k=0..1 give A000007, A384943.

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, 6*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,6*j)/j. Then A(n,k) = b(n,-k).