A383818 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - k*x) * Product_{j=0..k-1} (1 + j*x)/(1 - j*x).

1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 9, 10, 1, 0, 1, 16, 45, 22, 1, 0, 1, 25, 136, 177, 46, 1, 0, 1, 36, 325, 856, 621, 94, 1, 0, 1, 49, 666, 3025, 4576, 2049, 190, 1, 0, 1, 64, 1225, 8646, 23125, 22216, 6525, 382, 1, 0, 1, 81, 2080, 21217, 90126, 156145, 101536, 20337, 766, 1, 0

Offset: 0

Author

Seiichi Manyama, May 14 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1, 1,  1,    1,     1,      1, ...
  0, 1,  4,    9,    16,     25, ...
  0, 1, 10,   45,   136,    325, ...
  0, 1, 22,  177,   856,   3025, ...
  0, 1, 46,  621,  4576,  23125, ...
  0, 1, 94, 2049, 22216, 156145, ...
		

Crossrefs

Columns k=0..4 give A000007, A000012, A033484, A383912, A383913.

Main diagonal gives A383839.

Cf. A287532, A383900.

Programs

PARI

a(n, k) = sum(j=0, k, abs(stirling(k, j, 1))*stirling(j+n, k, 2));

Formula

A(n,k) = Sum_{j=0..k} |Stirling1(k,j)| * Stirling2(j+n,k).