A263416 -id:A263416 - cp's OEIS frontend

A261024 Decimal expansion of Cl_2(2*Pi/3), where Cl_2 is the Clausen function of order 2.

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

Offset: 0

Jean-François Alcover, Aug 07 2015

			0.676627737606435750014135036183013523961126205020199861344992737851...

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), A261025 (Cl_2(Pi/4)), A261026 (Cl_2(3*Pi/4)), A261027 (Cl_2(Pi/6)), A261028 (Cl_2(5*Pi/6)).

Cf. A252798, A252799, A263416, A263417.

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

clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
clausen(2, 2*Pi/3) \\ Gheorghe Coserea, Sep 30 2018

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

A263414 Expansion of Product_{k>=1} 1/(1-x^(3*k+4))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 4, 2, 0, 5, 6, 1, 6, 10, 2, 7, 19, 6, 9, 28, 14, 11, 44, 28, 16, 61, 52, 25, 87, 93, 45, 116, 153, 77, 160, 244, 141, 215, 376, 244, 301, 560, 422, 422, 817, 695, 617, 1173, 1132, 917, 1661, 1776, 1399, 2331

Offset: 0

Vaclav Kotesovec, Oct 17 2015

nonn

In general, if v>0, GCD(v,3)=1 and g.f. = Product_{k>=1} 1/(1-x^(3*k+v))^k, then
a(n) ~ d3(v) * 3^(v^2/27 - 8/9) * exp(-Pi^4 * v^2 / (3888*Zeta(3)) - v * Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) * n^(v^2/54 - 25/36) / (sqrt(Pi) * 2^(v^2/54 + 11/36) * Zeta(3)^(v^2/54 - 7/36)), where
d3(v) = exp(Integral_{x=0..infinity} (exp((3-v)*x) / (exp(3*x)-1)^2 + (1/12 - v^2/18)/exp(x) - 1/(9*x^2) + v/(9*x))/x dx).
if mod(v,3)=1, then d3(v) = exp(A263031) * 2^((v+2)/6) * 3^((v+2)/18) * Pi^((v+2)/6) / (Gamma(1/3)^((v+2)/3) * A263416((v-1)/3)).
if mod(v,3)=2, then d3(v) = exp(A263030) * 2^((v+1)/6) * Pi^((v+1)/6) / (3^((v+1)/18) * Gamma(2/3)^((v+1)/3) * A263417((v-2)/3)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - The asymptotic ratio (80000 terms)

Cf. A262877, A262876, A263405 (v=1), A263406 (v=2), A263415 (v=5), A263031, A263416.

with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
       add(add(`if`(irem(d-3, 3, 'r')=1, d*r, 0)
       , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Oct 17 2015

nmax = 80; CoefficientList[Series[Product[1/(1-x^(3*k+4))^k,{k,1,nmax}],{x,0,nmax}],x]
nmax = 80; CoefficientList[Series[E^Sum[x^(7*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(7*k)/(k*(1-x^(3*k))^2)).

a(n) ~ c * exp(-Pi^4/(243*Zeta(3)) - 4*Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (sqrt(Pi) * 2^(65/108) * 3^(8/27) * Zeta(3)^(11/108) * n^(43/108)), where c = exp(A263031) * 2 * 3^(1/3) * Pi / Gamma(1/3)^2 = 1.24446091929106216111829684663735422946506...

A263415 Expansion of Product_{k>=1} 1/(1-x^(3*k+5))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 1, 4, 0, 2, 5, 0, 6, 6, 1, 10, 7, 2, 19, 8, 6, 28, 10, 14, 44, 12, 28, 60, 17, 52, 86, 26, 93, 112, 46, 152, 152, 78, 243, 196, 142, 372, 264, 244, 552, 350, 422, 798, 486, 692, 1136, 680, 1125, 1582, 997, 1758

Offset: 0

Vaclav Kotesovec, Oct 17 2015

nonn

In general, if v>0, GCD(v,3)=1 and g.f. = Product_{k>=1} 1/(1-x^(3*k+v))^k, then
a(n) ~ d3(v) * 3^(v^2/27 - 8/9) * exp(-Pi^4 * v^2 / (3888*Zeta(3)) - v * Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) * n^(v^2/54 - 25/36) / (sqrt(Pi) * 2^(v^2/54 + 11/36) * Zeta(3)^(v^2/54 - 7/36)), where
d3(v) = exp(Integral_{x=0..infinity} (exp((3-v)*x) / (exp(3*x)-1)^2 + (1/12 - v^2/18)/exp(x) - 1/(9*x^2) + v/(9*x))/x dx).
if mod(v,3)=1, then d3(v) = exp(A263031) * 2^((v+2)/6) * 3^((v+2)/18) * Pi^((v+2)/6) / (Gamma(1/3)^((v+2)/3) * A263416((v-1)/3)).
if mod(v,3)=2, then d3(v) = exp(A263030) * 2^((v+1)/6) * Pi^((v+1)/6) / (3^((v+1)/18) * Gamma(2/3)^((v+1)/3) * A263417((v-2)/3)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - The asymptotic ratio (80000 terms)

Cf. A262877, A262876, A263405 (v=1), A263406 (v=2), A263414 (v=4), A263030, A263417.

with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
       add(add(`if`(irem(d-3, 3, 'r')=2, d*r, 0)
        , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Oct 17 2015

nmax = 80; CoefficientList[Series[Product[1/(1-x^(3*k+5))^k,{k,1,nmax}],{x,0,nmax}],x]
nmax = 80; CoefficientList[Series[E^Sum[x^(8*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(8*k)/(k*(1-x^(3*k))^2)).

a(n) ~ c * 3^(1/27) * exp(-25*Pi^4 / (3888*Zeta(3)) - 5*Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (sqrt(Pi) * 2^(83/108) * Zeta(3)^(29/108) * n^(25/108)), where c = exp(A263030) * Pi / (3^(1/3) * Gamma(2/3)^2) = 0.98365214791227284535715328899346961376609...

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

A369468 a(n) = Product_{k=0..n} ((3k+1)(3*k+2))^(n-k).

Original entry on oeis.org

1, 2, 80, 179200, 44154880000, 1980116762624000000, 24153039733453645414400000000, 111953168097640511435244254003200000000000, 262573865013264352348221085395200893360537600000000000000, 400294812944619753243237971399105071635747117771700305920000000000000000000

Offset: 0

Vaclav Kotesovec, Jan 23 2024

nonn

Cf. A263416, A263417.

Cf. A074962, A168440, A169619, A169620.

Table[Product[((3*k+1)*(3*k+2))^(n-k), {k, 0, n}], {n, 0, 10}]
Round[Table[3^(n^2 + 3*n/2 + 1/2) * BarnesG[n + 4/3] * BarnesG[n + 5/3] / (BarnesG[1/3] * BarnesG[2/3] * (2*Pi)^(n+1)), {n, 0, 10}]]
Round[Table[Glaisher^(8/3) * Gamma[1/3]^(1/3) * BarnesG[n + 4/3] * BarnesG[n + 5/3] * 3^(n^2 + 3*n/2 + 11/36) / (Exp[2/9] * (2*Pi)^(n + 2/3)), {n, 0, 10}]]

a(n) ~ A^(2/3) * Gamma(1/3)^(1/3) * 3^(n^2 + 3*n/2 + 11/36) * n^(n^2 + n + 1/9) / ((2*Pi)^(1/6) * exp(3*n^2/2 + n + 1/18)), where A is the Glaisher-Kinkelin constant A074962.
a(n) = A263416(n) * A263417(n).
a(n) = 3^(n^2 + 3*n/2 + 1/2) * BarnesG(n + 4/3) * BarnesG(n + 5/3) / (BarnesG(1/3) * BarnesG(2/3) * (2*Pi)^(n+1)).

A263430 a(n) = Product_{k=0..n} (4*k+1)^(n-k).

Original entry on oeis.org

1, 1, 5, 225, 131625, 1309010625, 273380323978125, 1427352844030287890625, 216119240915841469025244140625, 1079864992142473709995957417730712890625, 199639840782299404795675492100337942688751220703125

Offset: 0

Vaclav Kotesovec, Oct 18 2015

nonn

G. C. Greubel, Table of n, a(n) for n = 0..36
Eric Weisstein's World of Mathematics, Catalan's Constant
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant

Cf. A000178, A057863, A263416.

[(&*[(4*k+1)^(n-k): k in [0..n]]): n in [0..10]]; // G. C. Greubel, Aug 25 2018

Table[Product[(4*k+1)^(n-k),{k,0,n}],{n,0,10}]

for(n=0,10, print1(prod(k=0,n, (4*k+1)^(n-k)), ", ")) \\ G. C. Greubel, Aug 25 2018

a(n) ~ A^(1/8) * 2^(n^2 + 3*n/2 + 1/8) * Pi^(n/2 + 1/8) * n^(n^2/2 + n/4 - 5/96) / (Gamma(1/4)^(n + 1/4) * exp(3*n^2/4 + n/4 + 1/96 - C/(4*Pi))), where A = A074962 is the Glaisher-Kinkelin constant and C = A006752 = is Catalan's constant.

cp's OEIS Frontend

A261024 Decimal expansion of Cl_2(2*Pi/3), where Cl_2 is the Clausen function of order 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A263414 Expansion of Product_{k>=1} 1/(1-x^(3*k+4))^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A263415 Expansion of Product_{k>=1} 1/(1-x^(3*k+5))^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A369468 a(n) = Product_{k=0..n} ((3*k+1)*(3*k+2))^(n-k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A263430 a(n) = Product_{k=0..n} (4*k+1)^(n-k).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A369468 a(n) = Product_{k=0..n} ((3k+1)(3*k+2))^(n-k).