%I #23 Aug 02 2020 03:35:30

%S 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,

%T 32,113,197,132,51,1,1,7,47,234,688,903,429,127,1,1,8,65,424,1854,

%U 4404,4279,1430,323,1,1,9,86,699,4159,15490,29219,20793,4862,835

%N 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.

%H Seiichi Manyama, <a href="/A336706/b336706.txt">Antidiagonals n = 0..139, flattened</a>

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

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 3, 4, 5, 6, 7, ...

%e 2, 5, 11, 20, 32, 47, 65, ...

%e 4, 14, 45, 113, 234, 424, 699, ...

%e 9, 42, 197, 688, 1854, 4159, 8192, ...

%e 21, 132, 903, 4404, 15490, 43097, 101538, ...

%t 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 *)

%o (PARI) {T(n, k) = if(n==0, 1, sum(j=1, n, binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}

%o (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)}

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

%Y Main diagonal gives A335871.

%Y Cf. A336575, A336707, A336708, A336709.

%K nonn,tabl

%O 0,9

%A _Seiichi Manyama_, Aug 01 2020