A261024 -id:A261024 - cp's OEIS frontend

A261027 Decimal expansion of Cl_2(Pi/6), where Cl_2 is the Clausen function of order 2.

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

Offset: 0

Jean-François Alcover, Aug 07 2015

			0.8643791310538927496250363151902194786681885764036897041820...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Clausen Function
Eric Weisstein's MathWorld, Clausen's Integral
Eric Weisstein's MathWorld, Barnes G-Function
Wikipedia, Clausen function
Wikipedia, Barnes G-function

Cf. A006752 (Cl_2(Pi/2) = Catalan's constant), A143298 (Cl_2(Pi/3) = Gieseking's constant), A261024 (Cl_2(2*Pi/3)), A261025 (Cl_2(Pi/4)), A261026 (Cl_2(3*Pi/4)), A261028 (Cl_2(5*Pi/6)).

Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); RealDigits[Cl2[Pi/6] // Re, 10, 104] // First

Equals 2*Pi*log(G(11/12)/G(1/12)) - 2*Pi*LogGamma(1/12) + (Pi/6) * log(2*Pi*sqrt(2)/(sqrt(3)-1)), where G is the Barnes G function.

A261028 Decimal expansion of Cl_2(5*Pi/6), where Cl_2 is the Clausen function of order 2.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 07 2015

			0.356908327849065937114435038052959335697343922638539808173...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Clausen Function
Eric Weisstein's MathWorld, Clausen's Integral
Eric Weisstein's MathWorld, Barnes G-Function
Wikipedia, Clausen function
Wikipedia, Barnes G-function

Cf. A006752 (Cl_2(Pi/2) = Catalan's constant), A143298 (Cl_2(Pi/3) = Gieseking's constant), A261024 (Cl_2(2*Pi/3)), A261025 (Cl_2(Pi/4)), A261026 (Cl_2(3*Pi/4)), A261027 (Cl_2(Pi/6)).

Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); RealDigits[Cl2[5*Pi/6] // Re, 10, 104] // First

Equals 2*Pi*log(G(7/12)/G(5/12)) - 2*Pi*LogGamma(5/12) + (5*Pi/6) * log(2*Pi*sqrt(2)/(sqrt(3)+1)), where G is the Barnes G function.

A263416 a(n) = Product_{k=0..n} (3*k+1)^(n-k).

Original entry on oeis.org

1, 1, 4, 112, 31360, 114150400, 6648119296000, 7356542888181760000, 179090034163620983603200000, 108995627512253039588776345600000000, 1857397104331364341705287836001894400000000000, 981210407605679794692064339146706741991833600000000000000

Offset: 0

Vaclav Kotesovec, Oct 17 2015

G. C. Greubel, Table of n, a(n) for n = 0..50
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant
Eric Weisstein's World of Mathematics, Polygamma Function
Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function.

Cf. A057863, A074962, A252798, A261024, A263405, A263414, A263417, A369468.

A263416:=n->mul((3*k+1)^(n-k), k=0..n): seq(A263416(n), n=0..11); # Wesley Ivan Hurt, Nov 12 2015

Table[Product[(3*k+1)^(n-k),{k,0,n}],{n,0,12}] (* or *)
Table[1/FullSimplify[(Gamma[1/3]^((v-1)/3) / 3^((v-1)/18)) * Exp[Integrate[(E^((3-v)*x) - E^(2*x))/(x*(E^(3*x)-1)^2) + (v-1) * (1/(3*x*(E^(3*x)-1)) + 1/(6*x*E^(3*x)) - (v+1)/(18*x*E^x)), {x, 0, Infinity}]]], {v, 1, 34, 3}]
Round@Table[3^(n(n+1)/2) BarnesG[n+4/3]/(BarnesG[1/3] Gamma[1/3]^(n+1)), {n, 0, 12}] (* Vladimir Reshetnikov, Nov 11 2015 *)

a(n) = prod(k=0, n, (3*k+1)^(n-k)); \\ Michel Marcus, Nov 12 2015

a(n) ~ A^(1/3) * 2^(n/2 + 1/6) * 3^(n^2/2 + n/2 - 1/72) * n^(n^2/2 + n/3 - 1/36) * Pi^(n/2 + 1/6) / (Gamma(1/3)^(n + 1/3) * exp(3*n^2/4 + n/3 + Pi/(18*sqrt(3)) - PolyGamma(1, 1/3) / (12*sqrt(3)*Pi) + 1/36)), where A = A074962 is the Glaisher-Kinkelin constant and PolyGamma(1, 1/3) = 10.09559712542709408179200409989... (PolyGamma[1, 1/3] in Mathematica or Psi(1, 1/3) in Maple).
PolyGamma(1, 1/3) = 3^(3/2) * A261024 + 2*Pi^2/3.
From Vladimir Reshetnikov, Nov 11 2015: (Start)
a(n) = 3^(n*(n+1)/2) * G(n+4/3) / (G(1/3) * Gamma(1/3)^(n+1)), where G(x) is the Barnes G-function.
a(n) ~ 3^(n*(n+1)/2) * exp(-(9*n^2+4*n-1)/12) * n^((18*n^2+12*n-1)/36) * (2*Pi)^((3*n+1)/6) / (A * G(1/3) * Gamma(1/3)^(n+1)).
Note that G(1/3) = 3^(1/72) * exp(1/9 + Pi/(18*sqrt(3)) - PolyGamma(1, 1/3)/(12*sqrt(3)*Pi)) / (A^(4/3) * Gamma(1/3)^(2/3)).
(End)

A261025 Decimal expansion of Cl_2(Pi/4), where Cl_2 is the Clausen function of order 2.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 07 2015

			0.9818721510502033567179245430601956671307909716607304615766131...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's MathWorld, Clausen Function
Eric Weisstein's MathWorld, Clausen's Integral
Eric Weisstein's MathWorld, Barnes G-Function
Wikipedia, Clausen function
Wikipedia, Barnes G-function

Cf. A006752 (Cl_2(Pi/2) = Catalan's constant), A143298 (Cl_2(Pi/3) = Gieseking's constant), A261024 (Cl_2(2*Pi/3)), A261026 (Cl_2(3*Pi/4)), A261027 (Cl_2(Pi/6)), A261028 (Cl_2(5*Pi/6)).

Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); RealDigits[Cl2[Pi/4] // Re, 10, 105] // First

Equals 2*Pi*log(G(7/8)/G(1/8)) - 2*Pi*LogGamma(1/8) + (Pi/4) * log(2*Pi/sqrt(2-sqrt(2))), where G is the Barnes G function.

A261026 Decimal expansion of Cl_2(3*Pi/4), where Cl_2 is the Clausen function of order 2.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 07 2015

			0.52388935396159384919062278559400361174371628451989439444336407484...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Clausen Function
Eric Weisstein's MathWorld, Clausen's Integral
Eric Weisstein's MathWorld, Barnes G-Function
Wikipedia, Clausen function
Wikipedia, Barnes G-function

Cf. A006752 (Cl_2(Pi/2) = Catalan's constant), A143298 (Cl_2(Pi/3) = Gieseking's constant), A261024 (Cl_2(2*Pi/3)), A261025 (Cl_2(Pi/4)), A261027 (Cl_2(Pi/6)), A261028 (Cl_2(5*Pi/6)).

Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); RealDigits[Cl2[3*Pi/4] // Re, 10, 105] // First

Equals 2*Pi*log(G(5/8)/G(3/8)) - 2*Pi*LogGamma(3/8) + (3*Pi/4) * log(2*Pi/sqrt(2+sqrt(2))), where G is the Barnes G function.

A263417 a(n) = Product_{k=0..n} (3*k+2)^(n-k).

Original entry on oeis.org

1, 2, 20, 1600, 1408000, 17346560000, 3633063526400000, 15218176499384320000000, 1466155647574283911168000000000, 3672576800382377947366110003200000000000, 266783946802402043703868836144710942720000000000000

Offset: 0

Vaclav Kotesovec, Oct 17 2015

nonn

Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant
Eric Weisstein's World of Mathematics, Polygamma Function

Cf. A263416, A263406, A263415, A369468.

Table[Product[(3*k+2)^(n-k),{k,0,n}],{n,0,12}]
(* or *)
Table[1/FullSimplify[Gamma[2/3]^((v-2)/3) * 3^((v-2)/18) * Exp[Integrate[(E^((3-v)*x) - E^x)/(x*(E^(3*x)-1)^2) + (v-2) * (1/(3*x*(E^(3*x)-1)) + 1/(6*x*E^(3*x)) - (v+2)/(18*x*E^x)), {x, 0, Infinity}]]], {v, 2, 35, 3}]
Table[3^(n*(n+1)/2) * BarnesG[n + 5/3] / (BarnesG[2/3] * Gamma[2/3]^(n+1)), {n, 0, 12}] // Round (* Vaclav Kotesovec, Jan 23 2024 *)

a(n) ~ A^(1/3) * 2^(n/2 + 1/3) * 3^(n^2/2 + n/2 - 1/72) * Pi^(n/2 + 1/3) * n^(n^2/2 + 2*n/3 + 5/36) / (Gamma(2/3)^(n + 2/3) * exp(3*n^2/4 + 2*n/3 - Pi/(18*sqrt(3)) + PolyGamma(1, 1/3) / (12*sqrt(3)*Pi) + 1/36)), where A = A074962 is the Glaisher-Kinkelin constant and PolyGamma(1, 1/3) = 10.095597125427094081792004... (PolyGamma[1, 1/3] in Mathematica or Psi(1, 1/3) in Maple).
PolyGamma(1, 1/3) = 3^(3/2) * A261024 + 2*Pi^2/3.
a(n) = 3^(n*(n+1)/2) * BarnesG(n + 5/3) / (BarnesG(2/3) * Gamma(2/3)^(n+1)). - Vaclav Kotesovec, Jan 23 2024

A340826 Decimal expansion of Cl_2(Pi/5), where Cl_2 is the Clausen function of order 2.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jan 23 2021

			0.9237551681005353087119860297930...

For links see A261024.

Cf. A006752 (Cl_2(Pi/2) = Catalan's constant), A143298 (Cl_2(Pi/3) = Gieseking's constant), A261025 (Cl_2(Pi/4)), A261026 (Cl_2(3*Pi/4)), A261027 (Cl_2(Pi/6)), A261028 (Cl_2(5*Pi/6)), A340628, A340629.

Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); RealDigits[Re[Cl2[Pi/5]], 10, 105] // First
N[Pi*(ArcCsch[2] + Log[2*Pi*BarnesG[9/10]^10 / BarnesG[11/10]^10])/5, 120] (* Vaclav Kotesovec, Jan 23 2021 *)

A = Cl_2(Pi/5).
B = Cl_2(2*Pi/5).
C = Cl_2(3*Pi/5).
D = Cl_2(4*Pi/5).
4*(A^2 + C^2) = 5*(B^2 + D^2).
B = 2*A - 2*D.
D = 2*B - 2*C.
2*C = 4*A - 5*D.
B = -D + sqrt(A*(2*C+D)+D^2).
B^2 + D^2 = 4*Pi^4/(325*A340628^2).
B^2 + D^2 = (13/1125)*A340629^2*Pi^4.
Equals Pi*(2*log(G(9/10) / G(11/10)) + log(Pi*(1+sqrt(5)))/5), where G is the Barnes G-function. - Vaclav Kotesovec, Jan 23 2021

cp's OEIS Frontend

A261027 Decimal expansion of Cl_2(Pi/6), where Cl_2 is the Clausen function of order 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A261028 Decimal expansion of Cl_2(5*Pi/6), where Cl_2 is the Clausen function of order 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A263416 a(n) = Product_{k=0..n} (3*k+1)^(n-k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A261025 Decimal expansion of Cl_2(Pi/4), where Cl_2 is the Clausen function of order 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A261026 Decimal expansion of Cl_2(3*Pi/4), where Cl_2 is the Clausen function of order 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A263417 a(n) = Product_{k=0..n} (3*k+2)^(n-k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A340826 Decimal expansion of Cl_2(Pi/5), where Cl_2 is the Clausen function of order 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula