A324589 a(n) = Product_{j=1..n, k=1..n} (1 + (j*k)^2).

1, 2, 850, 9541930000, 62954953875193006250000, 2232026314050243695025069057306526600000000, 2378738322196706013428557679949358718247570924314917636028125000000000

Offset: 0

Author

Vaclav Kotesovec, Mar 09 2019

Keywords

• nonn

Comments

Product_{j>=1, k>=1} (1 + 1/(j^3*k^3)) = 3.07044599622955113359633939413741321690850038945774000273914990604256664558...

Crossrefs

Cf. A306760, A324590.

Programs

Maple

a:= n-> mul(mul((i*j)^2+1, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Mathematica

Table[Product[j^2*k^2 + 1, {j, 1, n}, {k, 1, n}], {n, 1, 8}]
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}]]

Formula

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

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023