A268504 -id:A268504 - cp's OEIS frontend

A306651 a(n) = Product_{k=1..n} BarnesG(3*k).

Original entry on oeis.org

1, 288, 36118462464000, 240498631970530185123135341199360000000000

Offset: 1

Vaclav Kotesovec, Mar 03 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, A268504, A296608, A306635, A324993.

Table[Product[BarnesG[3*k], {k, 1, n}], {n, 1, 6}]
Round[Table[3^(15*n^2/4 - 7*n/12 - 1/4) * E^(Pi/(18*Sqrt[3]) - PolyGamma[1, 1/3]/(12*Sqrt[3]*Pi) - Zeta[3]/(3*Pi^2) + 1/6 + 3*n*(n + 1)*(2*n + 1)/8 + 3*PolyGamma[-3, n + 1] - (3/2)*Derivative[1, 0][Zeta][-2, n] + (1/6)*Derivative[1, 0][Zeta][-2, 3*n] + (7/2)*Derivative[1, 0][Zeta][-1, n + 1/3] + (5/2)*Derivative[1, 0][Zeta][-1, n + 2/3]) * BarnesG[3*n]^(n + 1) * BarnesG[n + 1/3] * Gamma[n]^(5*n/2 - 13/6) / (BarnesG[4/3] * BarnesG[n]^(5/2) * Gamma[n + 1/3]^(n - 1) * Gamma[3*n]^(3*n*(n + 1)/2 - 2/3) * Glaisher^(3*n + 5) * (2*Pi)^(3*(n + 1)^2/4) * n^(3*n^2/2)), {n, 1, 6}]] (* Vaclav Kotesovec, Mar 04 2019 *)

a(n) ~ (2*Pi)^(3*n^2/4 + n/4 + 1/6) * 3^(3*n^3/2 + 3*n^2/4 - n/3 - 13/72) * n^(3*n^3/2 + 3*n^2/4 - n/3 - 5/72) / (Gamma(1/3)^(1/3) * A^(n + 1/6) * exp(11*n^3/4 + 9*n^2/8 - 5*n/12 - Zeta(3)/(24*Pi^2) - 1/72)), where A is the Glaisher-Kinkelin constant A074962.
a(n) = Product_{k=1..n} (exp(-8*Zeta'(-1)) * 3^(9*k^2/2 - 3*k + 5/12) * (2*Pi)^(1 - 3*k) * Gamma(k)^2 * Gamma(k + 1/3) * (BarnesG(k) * BarnesG(k + 1/3) * BarnesG(k + 2/3))^3).
a(n) = a(n-1)*A296608(n). - R. J. Mathar, Jul 24 2025

A367567 a(n) = Product_{k=0..n} (3*k)! / k!^3.

Original entry on oeis.org

1, 6, 540, 907200, 31434480000, 23788231346880000, 408042767492495815680000, 162838835029822082951032012800000, 1541352909587869227178909850805190656000000, 351233376660297011570511252132131832794456064000000000, 1949695346852822356399298814748829537555898997004605685760000000000

Offset: 0

Vaclav Kotesovec, Nov 23 2023

nonn

Cf. A000178, A006480, A268504, A061719, A268196.

Cf. A007685, A367568, A367569, A367570, A367571.

Table[Product[(3*k)!/k!^3, {k, 0, n}], {n, 0, 10}]
Table[Product[Binomial[3*k,k] * Binomial[2*k,k], {k, 0, n}], {n, 0, 10}]

a(n) = Product_{k=0..n} binomial(3*k,k) * binomial(2*k,k).
a(n) = A268504(n) / A000178(n)^3.
a(n) = A268504(n) / A061719(n).
a(n) = A007685(n) * A268196(n).
a(n) ~ A^(8/3) * Gamma(1/3)^(1/3) * 3^(3*n^2/2 + 2*n + 11/36) * exp(n - 2/9) / (n^(n + 13/18) * (2*Pi)^(n + 7/6)), where A is the Glaisher-Kinkelin constant A074962.

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

A272096 a(n) = Product_{k=0..n} (k*n)!.

Original entry on oeis.org

1, 1, 48, 1567641600, 9698137182219213471744000000, 21488900044302744250061179567064173417691432878080000000000000000

Offset: 0

Vaclav Kotesovec, Apr 20 2016

The next term has 126 digits.

Cf. A000178, A098694, A268504, A268505, A268506, A271946.

Cf. A255322, A272093.

Table[Product[(k*n)!, {k, 0, n}], {n, 0, 6}]

a(n) ~ A^n * n^(1/4 + 13*n/12 + n^2 + n^3) * (2*Pi)^(1/4 + n/2) / exp(n*(2 + 2*n + 3*n^2)/4), where A = A074962 is the Glaisher-Kinkelin constant.

cp's OEIS Frontend

A306651 a(n) = Product_{k=1..n} BarnesG(3*k).

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A367567 a(n) = Product_{k=0..n} (3*k)! / k!^3.

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

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A272096 a(n) = Product_{k=0..n} (k*n)!.

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