A362125 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - x*(1+x)^k)^k.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 3, 0, 1, 4, 15, 18, 5, 0, 1, 5, 26, 55, 47, 8, 0, 1, 6, 40, 124, 198, 118, 13, 0, 1, 7, 57, 235, 571, 681, 290, 21, 0, 1, 8, 77, 398, 1320, 2500, 2263, 702, 34, 0, 1, 9, 100, 623, 2640, 7026, 10504, 7341, 1677, 55, 0

Offset: 0

Author

Seiichi Manyama, Apr 08 2023

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1, 1,   1,   1,    1,    1, ...
  0, 1,   2,   3,    4,    5, ...
  0, 2,   7,  15,   26,   40, ...
  0, 3,  18,  55,  124,  235, ...
  0, 5,  47, 198,  571, 1320, ...
  0, 8, 118, 681, 2500, 7026, ...
		

Crossrefs

Columns k=0..3 give A000007, A000045(n+1), A362126, A382614.

Main diagonal gives A362080.

Cf. A362078, A362079.

Programs

PARI

T(n, k) = sum(j=0, n, binomial(j+k-1, j)*binomial(k*j, n-j));

Formula

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