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A258408 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^(2*k)) dx.

%I A258408 #16 Jun 02 2015 19:05:33
%S A258408 5,7,7,3,3,2,1,2,0,1,8,3,9,7,9,7,0,5,5,5,2,5,4,6,9,6,2,0,1,5,9,0,4,1,
%T A258408 5,5,0,8,0,1,1,9,3,1,3,8,3,5,6,3,4,9,2,4,5,5,8,9,0,8,8,0,3,7,5,1,5,2,
%U A258408 5,2,1,6,4,5,1,9,8,7,7,8,1,3,5,0,6,3,7,1,0,7,0,0,0,0,0,7,1,5,4,0,9,7,8,4,7,8
%N A258408 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^(2*k)) dx.
%C A258408 In general, Integral_{x=0..1} Product_{k>=1} (1-x^(m*k)) dx = Sum_{n} (-1)^n / (m*n*(3*n-1)/2 + 1) is equal to
%C A258408 if 0<m<24: 8*sqrt(3)*Pi * sinh(Pi/6*sqrt(24/m-1)) /
%C A258408            (sqrt((24-m)*m) * (2*cosh(Pi/3*sqrt(24/m-1))-1))
%C A258408 if m = 24: Pi^2/(6*sqrt(3)) = A258414
%C A258408 if m > 24: 8*sqrt(3)*Pi*sin(Pi/6*sqrt(1-24/m)) /
%C A258408            (sqrt((m-24)*m) * (2*cos(Pi/3*sqrt(1-24/m))-1)).
%C A258408 Integral_{x=0..1} Product_{k=1..n} (1+x^(m*k)) dx, where m >= 1, is asymptotic to 2*(m+1)^(n+1)/(m*n^2).
%C A258408 Integral_{x=-1..1} Product_{k>=1} (1-x^(2*k)) dx = 8*Pi*sqrt(3/11) * sinh(sqrt(11)*Pi/6) / (2*cosh(sqrt(11)*Pi/3)-1) = 1.154664240367959678... . - _Vaclav Kotesovec_, Jun 02 2015
%H A258408 Vaclav Kotesovec, <a href="http://oeis.org/A258232/a258232_2.pdf">The integration of q-series</a>
%F A258408 Equals 4*Pi*sqrt(3/11) * sinh(sqrt(11)*Pi/6) / (2*cosh(sqrt(11)*Pi/3) - 1).
%e A258408 0.5773321201839797055525469620159041550801193138356349245589088...
%p A258408 evalf(4*Pi*sqrt(3/11) * sinh(sqrt(11)*Pi/6) / (2*cosh(sqrt(11)*Pi/3) - 1), 120);
%t A258408 RealDigits[4*Pi*Sqrt[3/11]*Sinh[Sqrt[11]*Pi/6] / (2*Cosh[Sqrt[11]*Pi/3] - 1),10,120][[1]]
%Y A258408 Cf. A258232, A258406, A258414.
%K A258408 nonn,cons
%O A258408 0,1
%A A258408 _Vaclav Kotesovec_, May 29 2015