A366313 - OEIS

%I #14 Oct 06 2023 18:31:33

%S 0,10,41600,805545000,48248012800000,6993773647152500000,

%T 2092947132921735168000000,1157435764584534017163490000000,

%U 1090228457517544945858327347200000000,1643200095810939801357184785754425000000000

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

%H Mathematics StackExchange, <a href="https://math.stackexchange.com/questions/292080/a-limit-with-infinite-product-lim-limits-n-to-infty-frac1n4-prod-i-12">A limit with infinite product</a>, 2013.

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

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

%t Table[Product[k^2 + n^2, {k, 0, 2*n}], {n, 0, 10}]

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

%o (PARI) a(n) = prod(k=0, 2*n, n^2 + k^2); \\ _Michel Marcus_, Oct 06 2023

%Y Cf. A272244, A105199.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Oct 06 2023