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A258407 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^3 dx.

%I A258407 #25 Aug 23 2025 09:56:55
%S A258407 1,9,6,8,8,0,6,1,5,3,1,4,5,8,8,9,7,5,3,5,3,3,5,1,3,5,8,4,7,6,9,6,6,6,
%T A258407 8,2,9,6,6,7,3,4,3,1,7,8,3,9,1,7,5,7,5,8,6,0,9,3,3,5,7,0,6,2,6,8,9,9,
%U A258407 0,1,5,1,1,1,1,0,5,6,2,0,9,2,2,2,9,0,5,1,0,6,0,2,7,8,3,7,4,5,6,7,3,5,4,1,8,3
%N A258407 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^3 dx.
%C A258407 In general, Integral_{x=0..1} Product_{k>=1} (1-x^(m*k))^3 dx = Sum_{n>=0} (-1)^n * (2*n+1) / (m*n*(n+1)/2 + 1) is equal to
%C A258407 if 0<m<8: 2*Pi / (m * cosh((Pi/2)*sqrt(8/m-1)))
%C A258407 if m = 8: Pi/4
%C A258407 if m > 8: 2*Pi / (m * cos((Pi/2)*sqrt(1-8/m)))
%C A258407 Special values: m=4: Pi/(2*cosh(Pi/2)), m=9: 4*Pi/(9*sqrt(3)).
%C A258407 ---
%C A258407 Integral_{x=-1..1} Product_{k>=1} (1-x^k)^3 dx = 2*Pi*(1 + sqrt(2) * cosh(sqrt(7)*Pi/4)) / cosh(sqrt(7)*Pi/2) = 1.32639350417409769439126... . - _Vaclav Kotesovec_, Jun 02 2015
%H A258407 Vaclav Kotesovec, <a href="http://oeis.org/A258232/a258232_2.pdf">The integration of q-series</a>
%F A258407 Equals 2*Pi/cosh(sqrt(7)*Pi/2).
%F A258407 Equals Sum_{n>=0} (-1)^n * (2*n+1) / (n*(n+1)/2 + 1).
%e A258407 0.1968806153145889753533513584769666829667343178391757586093357...
%p A258407 evalf(2*Pi/cosh(sqrt(7)*Pi/2), 120);
%p A258407 evalf(Sum((-1)^n * (2*n+1) / (n*(n+1)/2 + 1), n=0..infinity), 120);
%t A258407 RealDigits[2*Pi*Sech[(Sqrt[7]*Pi)/2],10,105][[1]]
%o A258407 (PARI) 2*Pi/cosh((sqrt(7)*Pi)/2) \\ _Stefano Spezia_, Aug 23 2025
%Y A258407 Cf. A010816, A258232, A258406, A258404, A258405.
%K A258407 nonn,cons,changed
%O A258407 0,2
%A A258407 _Vaclav Kotesovec_, May 29 2015