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A336709 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 A336709 #22 Aug 02 2020 03:35:25
%S A336709 1,1,1,1,1,-2,1,1,-1,2,1,1,0,-1,4,1,1,1,-1,5,-24,1,1,2,2,0,-3,48,1,1,
%T A336709 3,8,5,2,-21,24,1,1,4,17,36,13,0,51,-464,1,1,5,29,109,177,36,-5,41,
%U A336709 1376,1,1,6,44,240,766,922,104,0,-391,-704
%N A336709 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 A336709 Seiichi Manyama, <a href="/A336709/b336709.txt">Antidiagonals n = 0..139, flattened</a>
%F A336709 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 A336709 Square array begins:
%e A336709     1,   1,  1,  1,   1,    1,     1, ...
%e A336709     1,   1,  1,  1,   1,    1,     1, ...
%e A336709    -2,  -1,  0,  1,   2,    3,     4, ...
%e A336709     2,  -1, -1,  2,   8,   17,    29, ...
%e A336709     4,   5,  0,  5,  36,  109,   240, ...
%e A336709   -24,  -3,  2, 13, 177,  766,  2177, ...
%e A336709    48, -21,  0, 36, 922, 5699, 20910, ...
%t A336709 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 A336709 (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 A336709 (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 A336709 Columns k=0-3 give: A307969(n-1), (-1)^n * A154825(n), A090192, A246555.
%Y A336709 Main diagonal gives A336714.
%Y A336709 Cf. A336575, A336706, A336707, A336708.
%K A336709 sign,tabl
%O A336709 0,6
%A A336709 _Seiichi Manyama_, Aug 01 2020