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%I #25 Oct 01 2022 23:42:46
%S 3,8,7,5,7,8,4,5,8,5,0,3,7,4,7,7,5,2,1,9,3,4,5,3,9,3,8,3,3,8,7,6,7,4,
%T 4,0,0,2,7,8,1,6,1,0,7,0,7,3,5,6,3,8,4,6,1,7,6,8,0,6,7,2,6,2,9,7,5,7,
%U 9,9,3,6,4,6,8,3,2,1,3,2,5,4,6,9,5,8,3,7,6,2,9,0,7,5,3,6,0,7,7,4
%N Decimal expansion of Pi^3/8.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals Integral_{x>0} log(x)^2/(1+x^2) dx.
%F Equals Integral_{x=0..Pi/2} log(tan(x))^2 dx.
%F Equals Integral_{x=0..Pi/2} log(sin(x)^3)*log(sin(x))-(3*Pi/2)*log(2)^2 dx.
%F Equals 27/7 * sum_{k>=0} (binomial(2*k, k)/((2*k+1)^3*16^k);
%F Equals 27/7 * 4F3([1/2, 1/2, 1/2, 1/2], [3/2, 3/2, 3/2], 1/4), where pFq() is the generalized hypergeometric function.
%F From _Amiram Eldar_, Aug 21 2020: (Start)
%F Equals Integral_{x=0..oo} x^2/cosh(x) dx.
%F Equals 2 + Integral_{x=0..oo} x^2 * exp(-x) * tanh(x) dx. (End)
%F From _Gleb Koloskov_, Jun 15 2021: (Start)
%F Equals 2*Integral_{x=0..1} log(x)^2/(1+x^2) dx.
%F Equals 2*Integral_{x=1..oo} log(x)^2/(1+x^2) dx.
%F Equals 2*(-1)^n*Integral_{x=-1/e..0} W(n,x)*(1-W(n,x))*log(-W(n,x))^2/x/(1-W(n,x)^4) dx, where W=LambertW, for n=0 and n=-1. (End)
%e 3.875784585037477521934539383387674400278161070735638461768067262975799364683...
%t RealDigits[Pi^3/8, 10, 100][[1]]
%o (PARI) Pi^3/8 \\ _Charles R Greathouse IV_, Oct 01 2022
%Y Cf. A000796, A002388, A019669, A091476, A091925.
%K nonn,cons,easy
%O 1,1
%A _Jean-François Alcover_, Apr 24 2013
%E Offset corrected by _Rick L. Shepherd_, Jan 01 2014
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