A268505 a(n) = Product_{k=0..n} (4*k)!.

1, 24, 967680, 463520268288000, 9698137182219213471744000000, 23594617426193665303453830729600860160000000000, 14639242671589099207353038379393488170313478620292159897600000000000000

Offset: 0

Author

Vaclav Kotesovec, Apr 16 2016

Keywords

• nonn

Comments

Partial products of A100733. - Michel Marcus, Jul 06 2019

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..19

Crossrefs

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

Programs

Mathematica

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

PARI

{a(n) = prod(k=1, n, (4*k)!)} \\ Seiichi Manyama, Jul 06 2019

Formula

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.

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