A263417 -id:A263417 - 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.

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)

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...

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)).

cp's OEIS Frontend

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

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

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

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

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

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