A385020 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 A385016.

1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 11, 3, 0, 1, 4, 18, 16, -51, 0, 1, 5, 26, 40, -71, -190, 0, 1, 6, 35, 76, -45, -452, -401, 0, 1, 7, 45, 125, 43, -702, -1683, 3672, 0, 1, 8, 56, 188, 210, -836, -3784, 4336, 51925, 0, 1, 9, 68, 266, 474, -729, -6480, 144, 108645, 151539, 0

Offset: 0

Author

Seiichi Manyama, Jun 15 2025

Keywords

• sign
• tabl

Examples

			Square array begins:
  1,    1,     1,     1,     1,     1,      1, ...
  0,    1,     2,     3,     4,     5,      6, ...
  0,    5,    11,    18,    26,    35,     45, ...
  0,    3,    16,    40,    76,   125,    188, ...
  0,  -51,   -71,   -45,    43,   210,    474, ...
  0, -190,  -452,  -702,  -836,  -729,   -234, ...
  0, -401, -1683, -3784, -6480, -9360, -11800, ...
		

Crossrefs

Columns k=0..1 give A000007, A385016.

Programs

PARI

b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-n+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+j+k-1,j-1) * b(n-j,4*j)/j. Then A(n,k) = b(n,-k).