A296607 -id:A296607 - cp's OEIS frontend

A306635 a(n) = Product_{k=1..n} BarnesG(2*k).

Original entry on oeis.org

1, 2, 576, 14332723200, 72474629486854275072000000, 482580045081719158086051946616717605601280000000000000

Offset: 1

Vaclav Kotesovec, Mar 02 2019

nonn

Next term is too long to be included.

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

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

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 *)

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

A296608 a(n) = BarnesG(3*n).

Original entry on oeis.org

0, 1, 288, 125411328000, 6658606584104736522240000000, 792786697595796795607377086400871488552960000000000000

Offset: 0

Vaclav Kotesovec, Dec 16 2017

nonn

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

Cf. A000178, A268504, A296607, A306651.

Table[BarnesG[3*n], {n, 0, 10}]
Round[Table[Glaisher^8 * E^(-2/3) * 3^(9*n^2/2 - 3*n + 5/12) * (2*Pi)^(1 - 3*n) * BarnesG[n] * BarnesG[n + 1/3]^2 * BarnesG[n + 2/3]^3 * BarnesG[n + 1]^2 * BarnesG[n + 4/3], {n, 0, 10}]]

a(n) = A^8 * exp(-2/3) * 3^(9*n^2/2 - 3*n + 5/12) * (2*Pi)^(1 - 3*n) * BarnesG(n) * BarnesG(n + 1/3)^2 * BarnesG(n + 2/3)^3 * BarnesG(n+1)^2 * BarnesG(n + 4/3), where A is the Glaisher-Kinkelin constant A074962.
a(n) ~ 3^(9*n^2/2 - 3*n + 5/12) * n^(9*n^2/2 - 3*n + 5/12) * (2*Pi)^((3*n-1)/2) / (A * exp(27*n^2/4 - 3*n - 1/12)), where A is the Glaisher-Kinkelin constant A074962.
a(n) = A000178(3*n-2). - R. J. Mathar, Jul 24 2025

A296627 a(n) = BarnesG(4*n).

Original entry on oeis.org

0, 2, 24883200, 6658606584104736522240000000, 69113789582492712943486800506462734562847413501952000000000000000

Offset: 0

Vaclav Kotesovec, Dec 17 2017

nonn

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

Cf. A000178, A296607, A296608.

Table[BarnesG[4*n], {n, 0, 6}]
Round[Table[Glaisher^15 * E^(-5/4) * 2^(7/3 - 14*n + 16*n^2) * Pi^(3/2 - 6*n) * BarnesG[n] * BarnesG[1/4 + n]^2 * BarnesG[1/2 + n]^3 * BarnesG[3/4 + n]^4 * BarnesG[1 + n]^3 * BarnesG[5/4 + n]^2 * BarnesG[3/2 + n], {n, 0, 6}]]

a(n) = A^15 * exp(-5/4) * 2^(7/3 - 14*n + 16*n^2) * Pi^(3/2 - 6*n) * BarnesG(n) * BarnesG(1/4 + n)^2 * BarnesG(1/2 + n)^3 * BarnesG(3/4 + n)^4 * BarnesG(1 + n)^3 * BarnesG(5/4 + n)^2 * BarnesG(3/2 + n), where A is the Glaisher-Kinkelin constant A074962.

a(n) ~ 2^(16*n^2 - 6*n + 1/3) * n^(8*n^2 - 4*n + 5/12) * Pi^(2*n - 1/2) / (A * exp(12*n^2 - 4*n - 1/12)), where A is the Glaisher-Kinkelin constant A074962.

cp's OEIS Frontend

A306635 a(n) = Product_{k=1..n} BarnesG(2*k).

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A296608 a(n) = BarnesG(3*n).

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A296627 a(n) = BarnesG(4*n).

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