1,2

Next term is too long to be included.

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

Wikipedia, Barnes G-function

a(n) ~ c * 2^(2*n^3/3 + n^2/2 - n/4 - 3/8) * n^(2*n^3/3 - n/4) * Pi^(n^2/2 - 3/8) / (A^(n-2) * exp(11*n^3/9 - n/3 - Zeta(3)/(2*Pi^2) + 1/12)), where c = A255674^2 = 1.1446513373245340524595435844492841792576337833610236993... and A is the Glaisher-Kinkelin constant A074962.

a(n) ~ 2^(2*n^3/3 + n^2/2 - n/4 - 1/8) * n^(2*n^3/3 - n/4) * Pi^(n^2/2) / (A^n * exp(11*n^3/9 - n/3 - Zeta(3)/(16*Pi^2))), where A is the Glaisher-Kinkelin constant A074962.

a(n) = a(n-1)*A296607(n). - R. J. Mathar, Jul 24 2025

Table[Product[BarnesG[2*k], {k, 1, n}], {n, 1, 8}]

Round[Table[2^(2*n^3/3 + n^2 - 5*n/3 - 2/3) * E^(n^3/2 + 3*n^2/4 + n/4 + 1/12 - 3*Zeta[3]/(16*Pi^2) + 2*PolyGamma[-3, n + 1] + Derivative[1, 0][Zeta][-2, n + 1/2] + 2*Derivative[1, 0][Zeta][-1, n + 1/2]) * Gamma[n]^(2*n - 7/4) * BarnesG[2*n]^(n + 1) / (Glaisher^(2*n + 3) * Pi^(n^2/2 + n + 1/2) * n^(n^2) * Gamma[2*n]^(n^2 + n - 3/4) * BarnesG[n]^2), {n, 1, 8}]] (* Vaclav Kotesovec, Mar 04 2019 *)

Cf. A055462, A168467, A255674, A296607, A306651, A324992.

Sequence in context: A163277 A003830 A134371 * A212840 A129697 A214911

Adjacent sequences: A306632 A306633 A306634 * A306636 A306637 A306638

Vaclav Kotesovec, Mar 02 2019

approved