A383900 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of Product_{j=0..k} (1 + j*x)/(1 - j*x).

1, 1, 0, 1, 2, 0, 1, 6, 2, 0, 1, 12, 18, 2, 0, 1, 20, 72, 42, 2, 0, 1, 30, 200, 312, 90, 2, 0, 1, 42, 450, 1400, 1152, 186, 2, 0, 1, 56, 882, 4650, 8000, 3912, 378, 2, 0, 1, 72, 1568, 12642, 38250, 40520, 12672, 762, 2, 0, 1, 90, 2592, 29792, 142002, 271770, 190400, 39912, 1530, 2, 0

Offset: 0

Author

Seiichi Manyama, May 14 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1, 1,   1,    1,     1,      1, ...
  0, 2,   6,   12,    20,     30, ...
  0, 2,  18,   72,   200,    450, ...
  0, 2,  42,  312,  1400,   4650, ...
  0, 2,  90, 1152,  8000,  38250, ...
  0, 2, 186, 3912, 40520, 271770, ...
		

Crossrefs

Columns k=0..4 give A000007, A040000, A068293(n+1), A383910, A383911.

Main diagonal gives A350366.

A(n,n-1) gives A383767.

Cf. A383818, A383843.

Programs

PARI

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

Formula

A(n,k) = Sum_{j=0..k} |Stirling1(k+1,j+1)| * Stirling2(j+n,k).