A372140 a(n) = Product_{k=1..n} BarnesG(k)^k.

1, 1, 1, 1, 16, 3981312, 2271857773302207479808, 133781874275586180035265927852035878702421114880000000

Offset: 0

Author

Vaclav Kotesovec, Apr 20 2024

Keywords

• nonn

Comments

The next term has 113 digits.

Links

• Eric Weisstein's World of Mathematics, Barnes G-Function.
• Wikipedia, Barnes G-function.

Crossrefs

Cf. A055462, A255269.

Programs

Mathematica

Table[Product[BarnesG[k]^k, {k, 1, n}], {n, 0, 8}]

Formula

a(n) ~ (2*Pi)^(n*(n^2 - 1)/6) * n^(n^4/8 - n^3/12 - n^2/6 + n/24 + 19/720) / (A^(n^2/2 + n/2 - 1/3) * exp(7*n^4/32 - 5*n^3/72 - 7*n^2/24 - n/24 - zeta(3)/(8*Pi^2) + zeta'(-3)/6 + 23/720)), where A is the Glaisher-Kinkelin constant A074962, zeta(3) = A002117, zeta'(-3) = A259068.