A382347 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = [x^n] Product_{j=0..n} (1 + (k*n+j)*x).

1, 1, 1, 1, 3, 2, 1, 5, 26, 6, 1, 7, 74, 342, 24, 1, 9, 146, 1650, 5944, 120, 1, 11, 242, 4578, 48504, 127860, 720, 1, 13, 362, 9774, 189144, 1763100, 3272688, 5040, 1, 15, 506, 17886, 520024, 9660840, 76223664, 97053936, 40320, 1, 17, 674, 29562, 1164024, 34201080, 586813968, 3817038960, 3270729600, 362880

Offset: 0

Author

Seiichi Manyama, May 18 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
    1,      1,       1,       1,        1,        1, ...
    1,      3,       5,       7,        9,       11, ...
    2,     26,      74,     146,      242,      362, ...
    6,    342,    1650,    4578,     9774,    17886, ...
   24,   5944,   48504,  189144,   520024,  1164024, ...
  120, 127860, 1763100, 9660840, 34201080, 93638820, ...
		

Crossrefs

Columns k=0..3 give A000142, A384024, A383678, A382349.

Main diagonal gives A380707.

Cf. A165675.

Programs

PARI

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

Formula

A(n,k) = A165675((k+1)*n,k*n).

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

A(n,k) = (n+1)! * Sum_{j=0..n} (-1)^j * binomial(-k*n,j)/(n+1-j).

A(n,k) = ((k+1)*n)!/(k*n)! * (1 + k*n * Sum_{j=1..n} 1/(k*n+j)).