A268505 - OEIS

%I #23 Apr 23 2024 05:37:53

%S 1,24,967680,463520268288000,9698137182219213471744000000,

%T 23594617426193665303453830729600860160000000000,

%U 14639242671589099207353038379393488170313478620292159897600000000000000

%N a(n) = Product_{k=0..n} (4*k)!.

%C Partial products of A100733. - _Michel Marcus_, Jul 06 2019

%H Seiichi Manyama, <a href="/A268505/b268505.txt">Table of n, a(n) for n = 0..19</a>

%F a(n) ~ Gamma(1/4)^(1/2) * 2^(5/6 + 11*n/2 + 4*n^2) * exp(1/48 - 5*n/2 - 3*n^2) * n^(29/48 + 5*n/2 + 2*n^2) * Pi^(1/4 + n/2) / A^(1/4), where A = A074962 is the Glaisher-Kinkelin constant.

%F a(n) = A^(15/4) * sqrt(Gamma(1/4)) * exp(-5/16) * 2^(4*n^2 + 7*n/2 - 5/12) * BarnesG(n + 5/4) * BarnesG(n + 3/2) * BarnesG(n + 7/4) * BarnesG(n+2) / Pi^(3*n/2 + 1). - _Vaclav Kotesovec_, Apr 23 2024

%t Table[Product[(4*k)!,{k,0,n}],{n,0,8}]

%o (PARI) {a(n) = prod(k=1, n, (4*k)!)} \\ _Seiichi Manyama_, Jul 06 2019

%Y Cf. A000178, A098694, A100733, A268504, A268506, A271946, A271947.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Apr 16 2016