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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.

%I A336706 #23 Aug 02 2020 03:35:30
%S A336706 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 A336706 32,113,197,132,51,1,1,7,47,234,688,903,429,127,1,1,8,65,424,1854,
%U A336706 4404,4279,1430,323,1,1,9,86,699,4159,15490,29219,20793,4862,835
%N 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.
%H A336706 Seiichi Manyama, <a href="/A336706/b336706.txt">Antidiagonals n = 0..139, flattened</a>
%F A336706 G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 - x * A_k(x)).
%e A336706 Square array begins:
%e A336706    1,   1,   1,    1,     1,     1,      1, ...
%e A336706    1,   1,   1,    1,     1,     1,      1, ...
%e A336706    1,   2,   3,    4,     5,     6,      7, ...
%e A336706    2,   5,  11,   20,    32,    47,     65, ...
%e A336706    4,  14,  45,  113,   234,   424,    699, ...
%e A336706    9,  42, 197,  688,  1854,  4159,   8192, ...
%e A336706   21, 132, 903, 4404, 15490, 43097, 101538, ...
%t A336706 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 A336706 (PARI) {T(n, k) = if(n==0, 1, sum(j=1, n, binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}
%o A336706 (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 A336706 Columns k=0-3 give: A001006(n-1), A000108, A001003, A108447.
%Y A336706 Main diagonal gives A335871.
%Y A336706 Cf. A336575, A336707, A336708, A336709.
%K A336706 nonn,tabl
%O A336706 0,9
%A A336706 _Seiichi Manyama_, Aug 01 2020