A330843 Square array T(n,k) = [x^n] ((1+x)^(k+1) / (1-x)^(k-1))^n, n>=0, k>=0, read by descending antidiagonals.

1, 1, 0, 1, 2, -2, 1, 4, 6, 0, 1, 6, 30, 20, 6, 1, 8, 70, 256, 70, 0, 1, 10, 126, 924, 2310, 252, -20, 1, 12, 198, 2240, 12870, 21504, 924, 0, 1, 14, 286, 4420, 41990, 184756, 204204, 3432, 70, 1, 16, 390, 7680, 104006, 811008, 2704156, 1966080, 12870, 0

Offset: 0

Author

Seiichi Manyama, Feb 07 2020

Keywords

• sign
• tabl

Examples

			Square array begins:
    1,   1,     1,      1,      1,       1, ...
    0,   2,     4,      6,      8,      10, ...
   -2,   6,    30,     70,    126,     198, ...
    0,  20,   256,    924,   2240,    4420, ...
    6,  70,  2310,  12870,  41990,  104006, ...
    0, 252, 21504, 184756, 811008, 2521260, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened

Crossrefs

Columns k=1..7 give A000984, A091527, A001448, A262732, A211419, A262733, A211421.

Main diagonal is A332231.

Programs

Mathematica

T[n_, k_] := Sum[Binomial[(k + 1)*n, j] * Binomial[k*n - j - 1, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 05 2021 *)

Formula

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

T(n,k) = 1/n! * ((k+1)*n)!/Gamma(1 + (k+1)*n/2) * Gamma(1 + (k-1)*n/2)/((k-1)*n)!.