A324443 a(n) = Product_{i=1..n, j=1..n} (1 + i^2 + j^2).

1, 3, 972, 437987088, 1396064690700615936, 100943980553724942717460016640000, 408685260379151918936869901376463191556211834880000, 193581283410907012468703321819613695893448022144552623141894180044800000000

Offset: 0

Author

Vaclav Kotesovec, Feb 28 2019

Keywords

• nonn

Comments

Product_{i>=1, j>=1} (1 + 1/(i^2 + j^2)) is divergent.

Crossrefs

Cf. A101686, A324403, A324425, A324444, A306398.

Programs

Maple

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

Mathematica

Table[Product[1 + i^2 + j^2, {i, 1, n}, {j, 1, n}], {n, 1, 10}]

Formula

a(n) ~ c * 2^(n*(n+1)) * exp(Pi*n*(n+1)/2 - 3*n^2) * n^(2*n^2 + (Pi - 1)/2), where c = A306398 = 0.1740394919107672354475619059102344818913844938434521480869...

a(n) / A324403(n) ~ d * n^(Pi/2), where d = A306398 * 2^(3/4) * exp(-Pi/12) * Pi^(1/4) * Gamma(3/4) = 0.36753062884677326134620846786416595535234038999313...

Extensions

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