A185149 -id:A185149 - cp's OEIS frontend

A186252 a(n)=Product{k=0..n-1, (3k+1)*A000108(k)}.

1, 1, 4, 56, 2800, 509600, 342451200, 858867609600, 8105992499404800, 289789231853721600000, 39450746867638243737600000, 20541057076054410196318617600000, 41055903763279774965226732643942400000, 315984464183472044352469495097074620825600000

Offset: 0

Paul Barry, Feb 16 2011

a(n) is the determinant of the symmetric matrix (if(j<=floor((i+j)/2), A000984(j+1), A000984(i+1)))_{0<=i,j<=n}.
Wolfram Alpha suggests that
a(n) = -A^(3/2)*3^(n+1)*2^(n(n-1)+23/24)*Pi^((1-2n)/4)*G(n+1/2)*Gamma(n+1/3) /(4*e^(1/8)*Gamma(-2/3)*G(n+2)) where G is Barnes G-function, and A is the Glaisher Kinkelin constant.

Cf. A000108, A185149, A000984, A074962.

Table[Product[(3*k+1)*Binomial[2k,k]/(k+1),{k,0,n-1}],{n,0,10}] (* Vaclav Kotesovec, Nov 14 2014 *)

a(n) ~ A^(3/2) * 2^(n^2-n-7/24) * 3^n * exp(n/2-1/8) / (GAMMA(1/3) * Pi^(n/2) * n^(n/2+13/24)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 14 2014

cp's OEIS Frontend

A186252 a(n)=Product{k=0..n-1, (3k+1)*A000108(k)}.

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