A336708 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} (-1)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.
1, 1, 1, 1, 1, -1, 1, 1, 0, 0, 1, 1, 1, -1, 2, 1, 1, 2, 1, 0, -3, 1, 1, 3, 6, 1, 2, -1, 1, 1, 4, 14, 21, 1, 0, 11, 1, 1, 5, 25, 76, 80, 1, -5, -15, 1, 1, 6, 39, 182, 450, 322, 1, 0, -13, 1, 1, 7, 56, 355, 1447, 2818, 1347, 1, 14, 77, 1, 1, 8, 76, 611, 3532, 12175, 18352, 5798, 1, 0, -86
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, 1, ... -1, 0, 1, 2, 3, 4, 5, ... 0, -1, 1, 6, 14, 25, 39, ... 2, 0, 1, 21, 76, 182, 355, ... -3, 2, 1, 80, 450, 1447, 3532, ... -1, 0, 1, 322, 2818, 12175, 37206, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
Crossrefs
Programs
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Mathematica
T[0, k_] := 1; T[n_, k_] := Sum[(-1)^(n - j) * Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *) -
PARI
{T(n, k) = if(n==0, 1, sum(j=1, n, (-1)^(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+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 + x * A_k(x)).
Extensions
Typo in name corrected by Georg Fischer, Sep 19 2023