A258404 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^4 dx.
1, 6, 1, 8, 2, 0, 2, 4, 2, 2, 9, 4, 8, 5, 6, 5, 6, 1, 8, 0, 2, 6, 1, 3, 3, 4, 9, 8, 5, 7, 8, 6, 5, 3, 4, 3, 1, 3, 0, 6, 8, 5, 7, 8, 2, 8, 8, 0, 1, 8, 9, 9, 0, 3, 9, 8, 0, 4, 2, 9, 4, 5, 3, 5, 7, 9, 5, 3, 4, 1, 5, 3, 8, 0, 4, 3, 7, 1, 4, 8, 9, 6, 8, 8, 5, 3, 3, 7, 1, 2, 9, 9, 2, 1, 5, 8, 5, 4, 4, 8, 5, 2, 1, 8, 9, 9
Offset: 0
Examples
0.16182024229485656180261334985786534313068578288018990398...
Links
- Vaclav Kotesovec, The integration of q-series
Programs
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Maple
evalf(Sum((2*Pi*(-1)^m / cosh(sqrt(7 - 4*m + 12*m^2)*Pi/2)), m=-infinity..infinity), 120); # Vaclav Kotesovec, Dec 04 2015
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Mathematica
nmax=200; p=1; q4=Table[PrintTemporary[n]; p=Expand[p*(1-x^n)^4]; Total[CoefficientList[p,x]/Range[1,Exponent[p,x]+1]],{n,1,nmax}]; q4n=N[q4,1000]; Table[SequenceLimit[Take[q4n,j]],{j,Length[q4n]-100,Length[q4n],10}] NSum[2*(-1)^m*Pi/Cosh[Sqrt[7 - 4*m + 12*m^2]*Pi/2], {m, -Infinity, Infinity}, WorkingPrecision -> 120, NSumTerms -> 100] (* Vaclav Kotesovec, Dec 04 2015 *) RealDigits[NIntegrate[QPochhammer[x]^4, {x, 0, 1}, WorkingPrecision -> 120], 10, 106][[1]] (* Vaclav Kotesovec, Oct 10 2023 *) -
PARI
default(realprecision, 93); b(n) = cosh(sqrt(7 - 4*n + 12*n^2)*Pi/2); 2*Pi*(1/b(0) + sumalt(n=1, (-1)^n*(1/b(n) + 1/b(-n)))) \\ Gheorghe Coserea, Sep 26 2018
Formula
Sum_{m = -infinity..infinity} (2*Pi*(-1)^m / cosh(sqrt(7 - 4*m + 12*m^2)*Pi/2)). - Vaclav Kotesovec, Dec 04 2015
Extensions
More digits from Vaclav Kotesovec, Oct 10 2023