A366313 a(n) = Product_{k=0..2*n} (n^2 + k^2).

0, 10, 41600, 805545000, 48248012800000, 6993773647152500000, 2092947132921735168000000, 1157435764584534017163490000000, 1090228457517544945858327347200000000, 1643200095810939801357184785754425000000000

Offset: 0

Author

Vaclav Kotesovec, Oct 06 2023

Keywords

• nonn

Links

• Mathematics StackExchange, A limit with infinite product, 2013.

Crossrefs

Cf. A272244, A105199.

Programs

Mathematica

Table[Product[k^2 + n^2, {k, 0, 2*n}], {n, 0, 10}]
Table[n^2*Pochhammer[1 - I*n, 2*n]*Pochhammer[1 + I*n, 2*n], {n, 0, 10}]

PARI

a(n) = prod(k=0, 2*n, n^2 + k^2); \\ Michel Marcus, Oct 06 2023

Formula

a(n) = n * sinh(n*Pi) * Gamma(1 + (2-i)*n) * Gamma(1 + (2+i)*n)/Pi, where i is the imaginary unit.

a(n) ~ 5^(2*n + 1/2) * exp(2*n*(arctan(2) - 2)) * n^(4*n+2).