A258404 - OEIS

%I #24 Oct 10 2023 08:01:05

%S 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,

%T 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,

%U 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

%N Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^4 dx.

%H Vaclav Kotesovec, <a href="http://oeis.org/A258232/a258232_2.pdf">The integration of q-series</a>

%F Sum_{m = -infinity..infinity} (2*Pi*(-1)^m / cosh(sqrt(7 - 4*m + 12*m^2)*Pi/2)). - _Vaclav Kotesovec_, Dec 04 2015

%e 0.16182024229485656180261334985786534313068578288018990398...

%p 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

%t 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}]

%t 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 *)

%t RealDigits[NIntegrate[QPochhammer[x]^4, {x, 0, 1}, WorkingPrecision -> 120], 10, 106][[1]] (* _Vaclav Kotesovec_, Oct 10 2023 *)

%o (PARI) default(realprecision, 93);

%o b(n) = cosh(sqrt(7 - 4*n + 12*n^2)*Pi/2);

%o 2*Pi*(1/b(0) + sumalt(n=1, (-1)^n*(1/b(n) + 1/b(-n)))) \\ _Gheorghe Coserea_, Sep 26 2018

%Y Cf. A000727, A258232, A258406, A258407, A258404, A258405.

%K nonn,cons

%O 0,2

%A _Vaclav Kotesovec_, May 29 2015

%E More digits from _Vaclav Kotesovec_, Oct 10 2023