A087016 -id:A087016 - cp's OEIS frontend

A087013 Decimal expansion of G(1/4) where G is the Barnes G-function.

2, 9, 3, 7, 5, 5, 9, 6, 5, 3, 3, 8, 6, 0, 9, 9, 5, 4, 7, 1, 7, 6, 8, 1, 6, 1, 0, 3, 2, 0, 5, 4, 6, 1, 7, 6, 6, 2, 0, 6, 2, 5, 3, 5, 9, 6, 7, 9, 8, 4, 3, 0, 5, 0, 1, 4, 9, 5, 7, 8, 9, 8, 8, 6, 3, 3, 3, 9, 6, 0, 4, 3, 0, 4, 0, 8, 7, 5, 0, 2, 2, 7, 3, 6, 1, 0, 2, 7, 2, 4, 3, 3, 2, 7, 3, 7, 4, 8, 4, 9, 5, 7

Offset: 0

Eric W. Weisstein, Jul 30 2003

			0.29375...

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

Cf. A087014, A087015, A087016, A087017.

E^(3/32 - Catalan/(4*Pi))/(Glaisher^(9/8)*Gamma[1/4]^(3/4))
(* Or, since version 7.0, *) RealDigits[BarnesG[1/4], 10, 102] // First (* Jean-François Alcover, Jul 11 2014 *)

exp(9/8*zeta'(-1)-Catalan/4/Pi)/gamma(1/4)^(3/4) \\ Charles R Greathouse IV, Dec 12 2013

G(1/4) * G(3/4) = A087013 * A087015 = exp(3/16) / (A^(9/4) * 2^(1/8) * Pi^(1/4) * GAMMA(1/4)^(1/2)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015

A087014 Decimal expansion of G(1/2) where G is the Barnes G-function.

Original entry on oeis.org

6, 0, 3, 2, 4, 4, 2, 8, 1, 2, 0, 9, 4, 4, 6, 2, 0, 6, 1, 9, 1, 4, 2, 9, 2, 2, 4, 5, 3, 4, 7, 0, 2, 0, 7, 9, 8, 8, 3, 0, 0, 3, 4, 2, 0, 3, 8, 9, 4, 5, 9, 7, 6, 5, 3, 8, 7, 7, 6, 9, 2, 0, 4, 1, 1, 9, 4, 3, 2, 7, 8, 5, 6, 7, 9, 3, 3, 5, 2, 9, 0, 7, 4, 8, 2, 9, 8, 6, 8, 8, 3, 6, 9, 8, 7, 3, 7, 4, 1, 4, 5, 4

Offset: 0

Eric W. Weisstein, Jul 30 2003

			0.60324...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 136.

Junesang Choi, Certain classes of series involving the zeta function, J. Math. Anal. Applic. 231 (1999) 91-117.
Eric Weisstein's World of Mathematics, Barnes G-Function
Wikipedia, Barnes G-function

Cf. A087013, A087015, A087016, A087017, A074962.

(2^(1/24)*E^(1/8))/(Glaisher^(3/2)*Pi^(1/4))
(* Or, since version 7.0, *) RealDigits[BarnesG[1/2], 10, 102] // First (* Jean-François Alcover, Jul 11 2014 *)

2^(1/24)*exp(3/2*zeta'(-1))/Pi^(1/4) \\ Charles R Greathouse IV, Dec 12 2013

A087015 Decimal expansion of G(3/4) where G is the Barnes G-function.

Original entry on oeis.org

8, 4, 8, 7, 1, 7, 5, 7, 9, 7, 2, 3, 8, 9, 9, 2, 2, 8, 6, 0, 8, 2, 0, 7, 6, 1, 2, 2, 7, 7, 2, 2, 9, 9, 7, 2, 7, 6, 5, 5, 2, 2, 5, 4, 1, 3, 8, 4, 8, 6, 9, 3, 5, 6, 9, 6, 0, 3, 4, 4, 9, 4, 7, 4, 8, 7, 2, 8, 5, 5, 5, 0, 9, 9, 6, 3, 0, 9, 2, 5, 3, 9, 9, 7, 3, 4, 5, 2, 3, 7, 0, 3, 1, 5, 0, 2, 5, 9, 1, 4, 9, 8

Offset: 0

Eric W. Weisstein, Jul 30 2003

			0.84871...

Junesang Choi, H. M. Srivastava, Certain classes of series involving the Zeta Function, J. Math. Anal. Applic. 231 (1999) 91-117
Eric Weisstein's World of Mathematics, Barnes G-Function
Wikipedia, Barnes G-function

Cf. A087013, A087014, A087016, A087017.

E^(3/32 + Catalan/(4*Pi))/(Glaisher^(9/8)*Gamma[3/4]^(1/4))
(* Or, since version 7.0, *) RealDigits[BarnesG[3/4], 10, 102] // First (* Jean-François Alcover, Jul 11 2014 *)

exp(Catalan/4/Pi+9/8*zeta'(-1))/gamma(3/4)^(1/4) \\ Charles R Greathouse IV, Dec 12 2013

G(1/4) * G(3/4) = A087013 * A087015 = exp(3/16) / (A^(9/4) * 2^(1/8) * Pi^(1/4) * GAMMA(1/4)^(1/2)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015

A087017 Decimal expansion of G(5/2) where G is the Barnes G-function.

Original entry on oeis.org

9, 4, 7, 5, 7, 3, 9, 0, 1, 0, 8, 3, 8, 2, 5, 7, 7, 6, 8, 8, 4, 1, 5, 2, 9, 8, 6, 3, 5, 3, 4, 5, 8, 0, 6, 4, 3, 7, 6, 4, 1, 0, 2, 6, 2, 7, 2, 4, 3, 1, 3, 3, 4, 2, 2, 4, 0, 2, 6, 9, 8, 4, 9, 8, 5, 7, 6, 6, 1, 2, 3, 2, 6, 3, 2, 6, 9, 6, 9, 9, 2, 9, 5, 9, 9, 8, 3, 0, 4, 3, 9, 4, 4, 9, 3, 5, 8, 6, 2, 7, 5, 8

Offset: 0

Eric W. Weisstein, Jul 30 2003

			0.94757...

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

Cf. A087013, A087014, A087015, A087016.

(E^(1/8)*Pi^(3/4))/(2^(23/24)*Glaisher^(3/2))
(* Or, since version 7.0, *) RealDigits[BarnesG[5/2], 10, 102] // First (* Jean-François Alcover, Jul 11 2014 *)

Pi^(3/4)*exp(3/2*zeta'(-1))/2^(23/24) \\ Charles R Greathouse IV, Dec 12 2013

Equals A087016 * A019704. - R. J. Mathar, Jul 24 2025

A252799 Decimal expansion of G(2/3) where G is the Barnes G-function.

Original entry on oeis.org

7, 7, 6, 8, 4, 9, 3, 8, 5, 7, 7, 6, 1, 8, 1, 4, 7, 7, 3, 0, 1, 1, 8, 3, 4, 3, 9, 2, 2, 1, 5, 4, 9, 9, 8, 0, 8, 0, 4, 0, 4, 7, 1, 3, 6, 3, 4, 5, 3, 8, 1, 3, 0, 0, 6, 2, 4, 5, 8, 7, 5, 2, 3, 8, 8, 7, 3, 0, 1, 2, 8, 1, 2, 8, 5, 0, 0, 0, 6, 0, 4, 0, 0, 9, 7, 4, 7, 4, 3, 0, 3, 1, 4, 8, 2, 4, 3, 7, 0, 5, 7, 9, 6, 1, 6

Offset: 0

Jean-François Alcover, Dec 22 2014

			0.7768493857761814773011834392215499808040471363453813...

V. S. Adamchik, Contributions to the Theory of the Barnes function, arXiv:math/0308086 [math.CA], 2003.
Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function.

Cf. A074962, A087013, A087014, A087015, A087016, A087017, A252798.

RealDigits[BarnesG[2/3], 10, 105] // First

(3^(1/72)*e^(1/9 + (-2*Pi^2 + 3*PolyGamma(1, 1/3))/(36*sqrt(3)*Pi)))/(A^(4/3)*Gamma(2/3)^(1/3)), where PolyGamma(1, .) is the derivative of the digamma function and A the Glaisher-Kinkelin constant (A074962).

G(1/3) * G(2/3) = A252798 * A252799 = 3^(7/36) * exp(2/9) / (A^(8/3) * 2^(1/3) * Pi^(1/3) * Gamma(1/3)^(1/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015

A252798 Decimal expansion of G(1/3) where G is the Barnes G-function.

Original entry on oeis.org

4, 0, 0, 0, 7, 8, 5, 2, 3, 0, 9, 0, 7, 6, 8, 2, 0, 2, 2, 8, 5, 0, 1, 4, 5, 1, 5, 2, 6, 0, 3, 0, 4, 5, 5, 7, 9, 2, 3, 0, 3, 8, 6, 3, 0, 8, 2, 8, 4, 1, 7, 5, 9, 8, 5, 9, 5, 3, 3, 2, 7, 0, 6, 2, 1, 9, 0, 9, 3, 8, 8, 9, 0, 3, 7, 1, 4, 6, 0, 9, 2, 0, 9, 0, 7, 5, 2, 9, 6, 6, 9, 9, 4, 6, 0, 2, 9, 9, 0, 2, 6, 9, 5, 6, 5

Offset: 0

Jean-François Alcover, Dec 22 2014

			0.4000785230907682022850145152603045579230386308284...

V. S. Adamchik, Contributions to the Theory of the Barnes function, arXiv:math/0308086 [math.CA], 2003.
Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function.

Cf. A074962, A087013, A087014, A087015, A087016, A087017, A252799.

RealDigits[BarnesG[1/3], 10, 105] // First

(3^(1/72)*e^(1/9 + (2*Pi^2 - 3*PolyGamma(1, 1/3))/(36*sqrt(3)*Pi)))/(A^(4/3)*Gamma(1/3)^(2/3)), where PolyGamma(1, .) is the derivative of the digamma function and A the Glaisher-Kinkelin constant (A074962).

G(1/3) * G(2/3) = A252798 * A252799 = 3^(7/36) * exp(2/9) / (A^(8/3) * 2^(1/3) * Pi^(1/3) * Gamma(1/3)^(1/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015

A185149 a(n) = 3^n*A003046(n+1)/A002457(n).

Original entry on oeis.org

1, 1, 3, 27, 756, 68040, 20207880, 20228087880, 69422797604160, 828491666608045440, 34788365080871828025600, 5191328567558179408948185600, 2776779354844059467693477099212800, 5363460395055494624228658756213491712000

Offset: 0

Paul Barry, Feb 15 2011

a(n) is the determinant of the symmetric matrix (if(j<=floor((i+j)/2), A000108(j+1), A000108(i+1)))_{0<=i,j<=n}.

G. C. Greubel, Table of n, a(n) for n = 0..60

Cf. A000108, A000245, A074962.

Cf. A087014, A087016, A087017 (some values of the Barnes G-function).

Table[Product[3*(2*k+2)!/((k+3)!*k!),{k,0,n-1}],{n,0,10}] (* Vaclav Kotesovec, Nov 14 2014 *)

a(n) = Product_{k=0..(n-1)} (A000108(k+2) - A000108(k+1)).
a(n) = Product_{k=0..(n-1)} 3(k+1)*A000108(k+1)/(k+3).
a(n) = Product_{k=0..(n-1)} A000245(k+1).
a(n) = (A^(3/2) 2^(n(n+1))*2^(23/24)*3^n*Pi^(-1/4-n/2)*G(n+3/2)*Gamma(n+1)) /(e^(1/8)*G(n+4)), where G is Barnes G-function, and A is the Glaisher-Kinkelin constant (A074962) (reported by Wolfram Alpha).
a(n) ~ A^(3/2) * 2^(n^2+n+5/24) * 3^n * exp(3*n/2-1/8) / (n^(3*n/2+31/8) * Pi^(n/2+1)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 14 2014

A252850 Decimal expansion of G(1/6) where G is the Barnes G-function.

Original entry on oeis.org

1, 8, 8, 9, 0, 9, 9, 3, 1, 2, 0, 7, 9, 0, 4, 6, 7, 0, 7, 6, 3, 6, 1, 0, 4, 1, 1, 0, 8, 7, 6, 4, 1, 0, 5, 2, 4, 1, 7, 5, 2, 8, 3, 5, 1, 5, 3, 9, 3, 3, 1, 6, 8, 6, 1, 4, 1, 9, 0, 3, 7, 9, 1, 2, 5, 7, 5, 5, 7, 8, 1, 4, 2, 6, 8, 8, 8, 2, 7, 9, 3, 4, 6, 4, 9, 9, 6, 8, 0, 4, 8, 6, 1, 6, 5, 9, 5, 5, 5, 4, 2, 9, 0, 4, 7

Offset: 0

Jean-François Alcover, Dec 23 2014

			0.1889099312079046707636104110876410524175283515393316861419...

Victor S. Adamchik, Contributions to the Theory of the Barnes function, arXiv:math/0308086 [math.CA], 2003.
Junesang Choi, H. M. Srivastava, and Victor S. Adamchik, Multiple Gamma and Related Functions, Applied Mathematics and Computation, Volume 134, Issues 2-3, 25 January 2003, pp. 515-533, see p. 7.
Eric Weisstein's MathWorld, Barnes G-Function.
Wikipedia, Barnes G-function.

Cf. A074962, A087013, A087014, A087015, A087016, A087017, A252798, A252799, A252851.

RealDigits[BarnesG[1/6], 10, 105] // First

Equals e^(5/72 + Pi/(12*sqrt(3)) - PolyGamma(1, 1/3)/(8*sqrt(3)*Pi))/(2^(1/72)*3^(1/144)*(A*Gamma(1/6))^(5/6)), where PolyGamma(1, .) is the derivative of the digamma function and A the Glaisher-Kinkelin constant (A074962).

G(1/6) * G(5/6) = A252850 * A252851 = exp(5/36) / (A^(5/3) * 2^(7/36) * 3^(1/72) * Pi^(1/6) * Gamma(1/6)^(2/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015

A252851 Decimal expansion of G(5/6) where G is the Barnes G-function.

Original entry on oeis.org

9, 0, 9, 7, 9, 9, 1, 9, 5, 8, 8, 8, 5, 9, 4, 0, 0, 6, 0, 6, 1, 4, 8, 8, 4, 0, 7, 2, 4, 5, 5, 8, 4, 9, 6, 9, 2, 9, 7, 7, 4, 4, 9, 4, 6, 9, 8, 7, 7, 5, 4, 7, 1, 2, 1, 8, 0, 7, 1, 9, 4, 0, 9, 1, 4, 7, 6, 9, 6, 9, 1, 0, 7, 0, 9, 1, 3, 5, 7, 1, 7, 4, 7, 0, 6, 8, 1, 7, 4, 6, 0, 2, 1, 8, 6, 6, 5, 3, 9

Offset: 0

Jean-François Alcover, Dec 23 2014

			0.90979919588859400606148840724558496929774494698775471218...

Victor S. Adamchik, Contributions to the Theory of the Barnes function, arXiv:math/0308086 [math.CA], 2003.
Junesang Choi, H. M. Srivastava, and Victor S. Adamchik, Multiple Gamma and Related Functions, Applied Mathematics and Computation, Volume 134, Issues 2-3, 25 January 2003, Pages 515-533, see p. 7.
Eric Weisstein's MathWorld, Barnes G-Function.
Wikipedia, Barnes G-function.

Cf. A074962, A087013, A087014, A087015, A087016, A087017, A252798, A252799, A252850.

RealDigits[BarnesG[5/6], 10, 99] // First

Equals e^(5/72 - Pi/(12*sqrt(3)) + PolyGamma(1, 1/3)/(8*sqrt(3)*Pi))/(2^(1/72)*3^(1/144)*A^(5/6)*Gamma(5/6)^(1/6)), where PolyGamma(1, .) is the derivative of the digamma function and A the Glaisher-Kinkelin constant (A074962).

G(1/6) * G(5/6) = A252850 * A252851 = exp(5/36) / (A^(5/3) * 2^(7/36) * 3^(1/72) * Pi^(1/6) * Gamma(1/6)^(2/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015

A256717 Decimal expansion of G(5/4) where G is the Barnes G-function.

Original entry on oeis.org

1, 0, 6, 5, 0, 4, 4, 5, 3, 8, 5, 3, 0, 9, 5, 5, 7, 1, 7, 1, 5, 9, 7, 1, 7, 5, 8, 3, 6, 9, 4, 9, 7, 7, 1, 4, 1, 9, 3, 7, 3, 4, 9, 0, 7, 3, 2, 6, 9, 7, 6, 1, 8, 9, 2, 2, 2, 1, 3, 9, 9, 3, 1, 5, 2, 0, 0, 4, 3, 8, 3, 7, 6, 1, 6, 8, 6, 0, 2, 2, 4, 4, 7, 6, 4, 6, 1, 5, 2, 5, 1, 0, 9, 9, 2, 8, 1, 4, 9, 1, 9, 4, 2, 3

Offset: 1

Jean-François Alcover, Apr 09 2015

			1.0650445385309557171597175836949771419373490732697618922213...

G. C. Greubel, Table of n, a(n) for n = 1..10000
J.-P. Allouche, A note on products involving zeta(3) and Catalan's constant. arXiv:1305.6247v3 [math.NT], 2013-2014, p. 7.
Eric Weisstein's MathWorld, Barnes G-Function
Wikipedia, Barnes G-function

Cf. A006752 (Catalan), A068466 (Gamma(1/4)), A074962 (Glaisher), A087013 (G(1/4)), A087014 (G(1/2)), A087015 (G(3/4)), A087016 (G(3/2)), A087017 (G(5/2)).

RealDigits[BarnesG[5/4], 10, 104] // First
RealDigits[Exp[3/32 - Catalan/(4*Pi)]*Gamma[1/4]^(1/4)/Glaisher^(9/8), 10, 100][[1]] (* G. C. Greubel, Aug 25 2018 *)

exp(3/32 - Catalan/(4*Pi))*gamma(1/4)^(1/4)/exp(3/32-9*zeta'(-1)/8) \\ Charles R Greathouse IV, Jul 01 2016

Equals exp(3/32 - Catalan/(4*Pi))*Gamma(1/4)^(1/4)/Glaisher^(9/8).

Equals G(1/4)*Gamma(1/4). - Vaclav Kotesovec, Apr 09 2015

cp's OEIS Frontend

A087013 Decimal expansion of G(1/4) where G is the Barnes G-function.

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A087014 Decimal expansion of G(1/2) where G is the Barnes G-function.

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A087015 Decimal expansion of G(3/4) where G is the Barnes G-function.

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A087017 Decimal expansion of G(5/2) where G is the Barnes G-function.

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A252799 Decimal expansion of G(2/3) where G is the Barnes G-function.

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A252798 Decimal expansion of G(1/3) where G is the Barnes G-function.

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A185149 a(n) = 3^n*A003046(n+1)/A002457(n).

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A252850 Decimal expansion of G(1/6) where G is the Barnes G-function.

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