A385019 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 A385015.

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 4, 0, 1, 4, 15, 16, -13, 0, 1, 5, 22, 37, -2, -81, 0, 1, 6, 30, 68, 45, -156, -389, 0, 1, 7, 39, 110, 141, -165, -1028, -198, 0, 1, 8, 49, 164, 300, -32, -1796, -1926, 7455, 0, 1, 9, 60, 231, 537, 336, -2460, -5499, 10923, 44515, 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,    4,     9,    15,    22,    30,    39, ...
  0,    4,    16,    37,    68,   110,   164, ...
  0,  -13,    -2,    45,   141,   300,   537, ...
  0,  -81,  -156,  -165,   -32,   336,  1050, ...
  0, -389, -1028, -1796, -2460, -2655, -1863, ...
		

Crossrefs

Columns k=0..1 give A000007, A385015.

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