A268196 a(n) = Product_{k=0..n} binomial(3*k,k).

1, 3, 45, 3780, 1871100, 5618913300, 104309506501200, 12129109415959536000, 8920608231265175901456000, 41809329673499408044341517200000, 1256161937180234817183361549396758000000, 243113461110708695347467432844366521953760000000

Offset: 0

Author

Vaclav Kotesovec, Apr 16 2016

Keywords

• nonn
• easy

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..49
• Eric Weisstein's World of Mathematics, Barnes G-Function.
• Wikipedia, Barnes G-function.

Crossrefs

Cf. A001142, A005809, A007685, A086205, A112332, A268504, A262261.

Programs

Mathematica

Table[Product[Binomial[3k,k],{k,0,n}],{n,0,12}]
FoldList[Times,Table[Binomial[3n,n],{n,0,15}]] (* Harvey P. Dale, Apr 23 2018 *)

Formula

a(n) = A^(7/6) * Gamma(1/3)^(1/3) * 3^(3*n^2/2 + 2*n + 11/36)* BarnesG(n + 4/3) * BarnesG(n + 5/3) / (exp(7/72) * 2^(n^2 + 2*n + 5/8) * Pi^(n/2 + 5/12) * BarnesG(n + 3/2) * BarnesG(n + 2)), where A = A074962 is the Glaisher-Kinkelin constant.

a(n) ~ A^(7/6) * Gamma(1/3)^(1/3) * 3^(11/36 + 2*n + 3*n^2/2) * exp(n/2 - 7/72) / (2^(n^2 + 2*n + 7/8) * Pi^(n/2 + 2/3) * n^(n/2 + 25/72)), where A = A074962 is the Glaisher-Kinkelin constant.

a(n) = A268504(n) / (A000178(n) * A098694(n)).