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A336707 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.

%I A336707 #23 Aug 02 2020 03:35:35
%S A336707 1,1,1,1,1,2,1,1,3,6,1,1,4,11,20,1,1,5,19,45,72,1,1,6,30,100,197,272,
%T A336707 1,1,7,44,201,562,903,1064,1,1,8,61,364,1445,3304,4279,4272,1,1,9,81,
%U A336707 605,3249,10900,20071,20793,17504,1,1,10,104,940,6502,30526,85128,124996,103049,72896
%N A336707 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.
%H A336707 Seiichi Manyama, <a href="/A336707/b336707.txt">Antidiagonals n = 0..139, flattened</a>
%F A336707 G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 - 2 * x * A_k(x)).
%e A336707 Square array begins:
%e A336707     1,   1,    1,     1,     1,     1,      1, ...
%e A336707     1,   1,    1,     1,     1,     1,      1, ...
%e A336707     2,   3,    4,     5,     6,     7,      8, ...
%e A336707     6,  11,   19,    30,    44,    61,     81, ...
%e A336707    20,  45,  100,   201,   364,   605,    940, ...
%e A336707    72, 197,  562,  1445,  3249,  6502,  11857, ...
%e A336707   272, 903, 3304, 10900, 30526, 73723, 158034, ...
%t A336707 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 *)
%o A336707 (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)}
%o A336707 (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)}
%Y A336707 Columns k=0-3 give: A071356(n-1), A001003, A007564, A118346.
%Y A336707 Main diagonal gives A336712.
%Y A336707 Cf. A336575, A336706, A336708, A336709.
%K A336707 nonn,tabl
%O A336707 0,6
%A A336707 _Seiichi Manyama_, Aug 01 2020