A369468 - OEIS

%I #12 Jan 23 2024 14:58:49

%S 1,2,80,179200,44154880000,1980116762624000000,

%T 24153039733453645414400000000,

%U 111953168097640511435244254003200000000000,262573865013264352348221085395200893360537600000000000000,400294812944619753243237971399105071635747117771700305920000000000000000000

%N a(n) = Product_{k=0..n} ((3*k+1)*(3*k+2))^(n-k).

%F a(n) ~ A^(2/3) * Gamma(1/3)^(1/3) * 3^(n^2 + 3*n/2 + 11/36) * n^(n^2 + n + 1/9) / ((2*Pi)^(1/6) * exp(3*n^2/2 + n + 1/18)), where A is the Glaisher-Kinkelin constant A074962.

%F a(n) = A263416(n) * A263417(n).

%F a(n) = 3^(n^2 + 3*n/2 + 1/2) * BarnesG(n + 4/3) * BarnesG(n + 5/3) / (BarnesG(1/3) * BarnesG(2/3) * (2*Pi)^(n+1)).

%t Table[Product[((3*k+1)*(3*k+2))^(n-k), {k, 0, n}], {n, 0, 10}]

%t Round[Table[3^(n^2 + 3*n/2 + 1/2) * BarnesG[n + 4/3] * BarnesG[n + 5/3] / (BarnesG[1/3] * BarnesG[2/3] * (2*Pi)^(n+1)), {n, 0, 10}]]

%t Round[Table[Glaisher^(8/3) * Gamma[1/3]^(1/3) * BarnesG[n + 4/3] * BarnesG[n + 5/3] * 3^(n^2 + 3*n/2 + 11/36) / (Exp[2/9] * (2*Pi)^(n + 2/3)), {n, 0, 10}]]

%Y Cf. A263416, A263417.

%Y Cf. A074962, A168440, A169619, A169620.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Jan 23 2024