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A336574 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j) * binomial(k*n+j+1,n)/(k*n+j+1).

%I A336574 #52 Aug 09 2025 16:48:40
%S A336574 1,1,3,1,3,6,1,3,15,12,1,3,24,93,24,1,3,33,255,645,48,1,3,42,498,3102,
%T A336574 4791,96,1,3,51,822,8691,40854,37275,192,1,3,60,1227,18708,164937,
%U A336574 566934,299865,384,1,3,69,1713,34449,464115,3305868,8164263,2474025,768
%N A336574 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j) * binomial(k*n+j+1,n)/(k*n+j+1).
%H A336574 Seiichi Manyama, <a href="/A336574/b336574.txt">Antidiagonals n = 0..139, flattened</a>
%F A336574 G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k * (1 + 2 * A_k(x)).
%F A336574 T(n,k) = (1/(k*n+1)) * Sum_{j=0..n} 2^(n-j) * binomial(k*n+1,j) * binomial((k+1)*n-j,n-j).
%F A336574 From _Seiichi Manyama_, Aug 10 2023: (Start)
%F A336574 T(n,k) = (1/n) * Sum_{j=0..n-1} (-1)^j * 3^(n-j) * binomial(n,j) * binomial((k+1)*n-j,n-1-j) for n > 0.
%F A336574 T(n,k) = (1/n) * Sum_{j=1..n} 3^j * 2^(n-j) * binomial(n,j) * binomial(k*n,j-1) for n > 0. (End)
%F A336574 T(n,k) = binomial(1+k*n, n)*hypergeom([-n, 1+k*n], [2+(k-1)*n], -2)/(1 + k*n) for k > 0. - _Stefano Spezia_, Aug 09 2025
%e A336574 Square array begins:
%e A336574    1,    1,     1,      1,      1,       1, ...
%e A336574    3,    3,     3,      3,      3,       3, ...
%e A336574    6,   15,    24,     33,     42,      51, ...
%e A336574   12,   93,   255,    498,    822,    1227, ...
%e A336574   24,  645,  3102,   8691,  18708,   34449, ...
%e A336574   48, 4791, 40854, 164937, 464115, 1055838, ...
%t A336574 T[n_, k_] := Sum[2^j * Binomial[n, j] * Binomial[k*n + j + 1, n]/(k*n + j + 1), {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Jul 27 2020 *)
%o A336574 (PARI) T(n, k) = sum(j=0, n, 2^j*binomial(n, j)*binomial(k*n+j+1, n)/(k*n+j+1));
%o A336574 (PARI) T(n, k) = my(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k*(1+2*A)); polcoeff(A, n);
%o A336574 (PARI) T(n, k) = sum(j=0, n, 2^(n-j)*binomial(k*n+1, j)*binomial((k+1)*n-j, n-j))/(k*n+1);
%Y A336574 Columns k=0-4 give: A003945, A103210, A219536, A336539, A336572.
%Y A336574 Main diagonal gives A336577.
%Y A336574 Cf. A336534, A336573, A336575.
%K A336574 nonn,tabl
%O A336574 0,3
%A A336574 _Seiichi Manyama_, Jul 26 2020