%I #13 Jun 24 2023 16:03:41
%S 1,2,850,9541930000,62954953875193006250000,
%T 2232026314050243695025069057306526600000000,
%U 2378738322196706013428557679949358718247570924314917636028125000000000
%N a(n) = Product_{j=1..n, k=1..n} (1 + (j*k)^2).
%C Product_{j>=1, k>=1} (1 + 1/(j^3*k^3)) = 3.07044599622955113359633939413741321690850038945774000273914990604256664558...
%F a(n) ~ c * 4^n * Pi^(2*n) * n^(2*n*(2*n+1)) / exp(4*n^2), where c = 14.2467190172413789737182639605567415110439648274273645215657580983939589... = exp(1/3) * Product_{j>=1, k>=1} (1 + 1/(j^2*k^2)). - _Vaclav Kotesovec_, Mar 28 2019
%p a:= n-> mul(mul((i*j)^2+1, i=1..n), j=1..n):
%p seq(a(n), n=0..7); # _Alois P. Heinz_, Jun 24 2023
%t Table[Product[j^2*k^2 + 1, {j, 1, n}, {k, 1, n}], {n, 1, 8}]
%t Round[Table[Product[k^(1 + 2*n) * Gamma[1 - I/k + n] * Gamma[1 + I/k + n] * Sinh[Pi/k]/Pi, {k, 1, n}], {n, 1, 8}]]
%Y Cf. A306760, A324590.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Mar 09 2019
%E a(0)=1 prepended by _Alois P. Heinz_, Jun 24 2023