A336706 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} binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 4, 1, 1, 4, 11, 14, 9, 1, 1, 5, 20, 45, 42, 21, 1, 1, 6, 32, 113, 197, 132, 51, 1, 1, 7, 47, 234, 688, 903, 429, 127, 1, 1, 8, 65, 424, 1854, 4404, 4279, 1430, 323, 1, 1, 9, 86, 699, 4159, 15490, 29219, 20793, 4862, 835

Offset

0,9

Links

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

Formula

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

Example

Square array begins:
   1,   1,   1,    1,     1,     1,      1, ...
   1,   1,   1,    1,     1,     1,      1, ...
   1,   2,   3,    4,     5,     6,      7, ...
   2,   5,  11,   20,    32,    47,     65, ...
   4,  14,  45,  113,   234,   424,    699, ...
   9,  42, 197,  688,  1854,  4159,   8192, ...
  21, 132, 903, 4404, 15490, 43097, 101538, ...

Mathematica

T[0, k_] := 1; T[n_, k_] := Sum[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 *)

Prog

(PARI) {T(n, k) = if(n==0, 1, sum(j=1, n, 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)}

Crossrefs

Columns k=0-3 give: A001006(n-1), A000108, A001003, A108447.

Main diagonal gives A335871.

Cf. A336575, A336707, A336708, A336709.

Sequence in context: A335845 A357611 A290252 * A344567 A076037 A215563

Adjacent sequences: A336703 A336704 A336705 * A336707 A336708 A336709

Keyword

nonn,tabl

Author

Seiichi Manyama, Aug 01 2020

Status

approved