A371603 - OEIS

%I #31 Apr 01 2024 08:38:04

%S 1,1,4,1134,333132800,1319947441510156250,

%T 876533819183888230348458418944000,

%U 1185269534290897564185384010731432113450477770983533184

%N a(n) = Product_{k=0..n} binomial(n^2, k^2).

%F a(n) = (n^2)!^(n+1) / (A255322(n) * A371624(n)).

%F a(n) ~ c * exp(2*n*(2*n^2/3 + 1)) / (A^(2*n) * 2^(4*n*(n^2 + 1)/3) * Pi^(n/2) * n^(7*n/6 - 1/4)), where c = 0.6367427... and A is the Glaisher-Kinkelin constant A074962.

%t Table[Product[Binomial[n^2, k^2], {k, 0, n}], {n, 0, 8}]

%Y Cf. A001142, A255322, A272095, A296591, A362288, A371624, A371646.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Mar 30 2024