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A336708 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} (-1)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

%I A336708 #28 Sep 19 2023 15:32:58
%S A336708 1,1,1,1,1,-1,1,1,0,0,1,1,1,-1,2,1,1,2,1,0,-3,1,1,3,6,1,2,-1,1,1,4,14,
%T A336708 21,1,0,11,1,1,5,25,76,80,1,-5,-15,1,1,6,39,182,450,322,1,0,-13,1,1,7,
%U A336708 56,355,1447,2818,1347,1,14,77,1,1,8,76,611,3532,12175,18352,5798,1,0,-86
%N A336708 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} (-1)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.
%H A336708 Seiichi Manyama, <a href="/A336708/b336708.txt">Antidiagonals n = 0..139, flattened</a>
%F A336708 G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 + x * A_k(x)).
%e A336708 Square array begins:
%e A336708    1,  1, 1,   1,    1,     1,     1, ...
%e A336708    1,  1, 1,   1,    1,     1,     1, ...
%e A336708   -1,  0, 1,   2,    3,     4,     5, ...
%e A336708    0, -1, 1,   6,   14,    25,    39, ...
%e A336708    2,  0, 1,  21,   76,   182,   355, ...
%e A336708   -3,  2, 1,  80,  450,  1447,  3532, ...
%e A336708   -1,  0, 1, 322, 2818, 12175, 37206, ...
%t A336708 T[0, k_] := 1; T[n_, k_] := Sum[(-1)^(n - j) * Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Aug 01 2020 *)
%o A336708 (PARI) {T(n, k) = if(n==0, 1, sum(j=1, n, (-1)^(n-j)*binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}
%o A336708 (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 A336708 Columns k=0-3 give: A007440, A090192, A000012, A106228.
%Y A336708 Main diagonal gives A336713.
%Y A336708 Cf. A336575, A336706, A336707, A336709.
%K A336708 sign,tabl
%O A336708 0,15
%A A336708 _Seiichi Manyama_, Aug 01 2020
%E A336708 Typo in name corrected by _Georg Fischer_, Sep 19 2023