A336709 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-2)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

1, 1, 1, 1, 1, -2, 1, 1, -1, 2, 1, 1, 0, -1, 4, 1, 1, 1, -1, 5, -24, 1, 1, 2, 2, 0, -3, 48, 1, 1, 3, 8, 5, 2, -21, 24, 1, 1, 4, 17, 36, 13, 0, 51, -464, 1, 1, 5, 29, 109, 177, 36, -5, 41, 1376, 1, 1, 6, 44, 240, 766, 922, 104, 0, -391, -704

Offset: 0

Author

Seiichi Manyama, Aug 01 2020

Keywords

• sign
• tabl

Examples

			Square array begins:
    1,   1,  1,  1,   1,    1,     1, ...
    1,   1,  1,  1,   1,    1,     1, ...
   -2,  -1,  0,  1,   2,    3,     4, ...
    2,  -1, -1,  2,   8,   17,    29, ...
    4,   5,  0,  5,  36,  109,   240, ...
  -24,  -3,  2, 13, 177,  766,  2177, ...
   48, -21,  0, 36, 922, 5699, 20910, ...
		

Links

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

Crossrefs

Columns k=0-3 give: A307969(n-1), (-1)^n * A154825(n), A090192, A246555.

Main diagonal gives A336714.

Cf. A336575, A336706, A336707, A336708.

Programs

Mathematica

T[0, k_] := 1; T[n_, k_] := Sum[(-2)^(n - j) * Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)

PARI

{T(n, k) = if(n==0, 1, sum(j=1, n, (-2)^(n-j)*binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}

PARI

{T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k/(1+2*x*A)); polcoef(A, n)}

Formula

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 + 2 * x * A_k(x)).