A384945 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 A384942.

1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 11, -5, 0, 1, 4, 18, 0, -135, 0, 1, 5, 26, 16, -255, -110, 0, 1, 6, 35, 44, -345, -540, 3661, 0, 1, 7, 45, 85, -389, -1230, 5777, 16440, 0, 1, 8, 56, 140, -370, -2100, 5918, 40452, -1375, 0, 1, 9, 68, 210, -270, -3049, 3784, 67356, 86065, -827075, 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,    5,   11,    18,    26,    35,    45, ...
  0,   -5,    0,    16,    44,    85,   140, ...
  0, -135, -255,  -345,  -389,  -370,  -270, ...
  0, -110, -540, -1230, -2100, -3049, -3954, ...
  0, 3661, 5777,  5918,  3784,  -770, -7708, ...
		

Crossrefs

Columns k=0..1 give A000007, A384942.

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, 5*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,5*j)/j. Then A(n,k) = b(n,-k).