A362078 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = [x^n] 1/(1 - x*(1+x)^k)^n.

1, 1, 1, 1, 1, 3, 1, 1, 5, 10, 1, 1, 7, 22, 35, 1, 1, 9, 37, 105, 126, 1, 1, 11, 55, 215, 511, 462, 1, 1, 13, 76, 369, 1271, 2534, 1716, 1, 1, 15, 100, 571, 2526, 7651, 12720, 6435, 1, 1, 17, 127, 825, 4401, 17577, 46614, 64449, 24310, 1, 1, 19, 157, 1135, 7026, 34412, 123810, 286599, 328900, 92378

Offset: 0

Author

Seiichi Manyama, Apr 08 2023

Keywords

• nonn
• tabl

Examples

			Square array begins:
    1,   1,    1,    1,    1,    1, ...
    1,   1,    1,    1,    1,    1, ...
    3,   5,    7,    9,   11,   13, ...
   10,  22,   37,   55,   76,  100, ...
   35, 105,  215,  369,  571,  825, ...
  126, 511, 1271, 2526, 4401, 7026, ...
		

Crossrefs

Columns k=0..3 give A088218, A213684, A362087, A362088.

Main diagonal gives A362080.

Cf. A362079.

Programs

PARI

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

Formula

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