A258407 - OEIS

%I #18 Jun 03 2015 03:21:20

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

%C 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 if 0<m<8: (2*Pi / (m * cosh(Pi/2*sqrt(8/m-1)))

%C if m = 8: Pi/4

%C if m > 8: (2*Pi / (m * cos(Pi/2*sqrt(1-8/m)))

%C Special values: m=4: Pi/(2*cosh(Pi/2)), m=9: 4*Pi/(9*sqrt(3)).

%C ---

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

%F Equals 2*Pi/cosh(sqrt(7)*Pi/2).

%F Equals Sum_{n>=0} (-1)^n * (2*n+1) / (n*(n+1)/2 + 1).

%e 0.1968806153145889753533513584769666829667343178391757586093357...

%p evalf(2*Pi/cosh(sqrt(7)*Pi/2), 120);

%p evalf(Sum((-1)^n * (2*n+1) / (n*(n+1)/2 + 1), n=0..infinity), 120);

%t RealDigits[2*Pi*Sech[(Sqrt[7]*Pi)/2],10,105][[1]]

%Y Cf. A010816, A258232, A258406, A258404, A258405.

%K nonn,cons

%O 0,2

%A _Vaclav Kotesovec_, May 29 2015