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A196878 Decimal expansion of (Pi/8)*(6*zeta(3)+Pi^2*log(2)+4*log(2)^3).

%I A196878 #26 Jun 12 2025 22:34:54
%S A196878 6,0,4,1,8,8,2,9,0,9,7,7,5,0,9,3,5,2,2,1,5,0,4,2,4,1,3,0,6,7,5,9,9,5,
%T A196878 9,8,5,5,0,8,7,1,0,3,0,5,7,7,4,6,4,1,9,0,7,2,5,8,6,0,1,0,1,5,2,6,0,0,
%U A196878 4,3,0,2,5,4,6,5,5,7,5,8,1,6,0,4,0,4,7,0,8,2,6,5,8,8,2,6,1,6,9,5,1,5,5,8,1
%N A196878 Decimal expansion of (Pi/8)*(6*zeta(3)+Pi^2*log(2)+4*log(2)^3).
%C A196878 The absolute value of the integral {x=0..Pi/2} log(sin(x))^3 dx. The absolute value of m=3 of sqrt(Pi)/2*(d^m/da^m(gamma((a+1)/2)/gamma(a/2+1))) at a=0. - _Seiichi Kirikami_ and _Peter J. C. Moses_, Oct 07 2011
%D A196878 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 3.621.1
%H A196878 G. C. Greubel, <a href="/A196878/b196878.txt">Table of n, a(n) for n = 1..5000</a>
%H A196878 K. S. Kolbig, <a href="https://doi.org/10.1090/S0025-5718-1983-0689472-3">On the integral int_0^Pi/2 log^n cos x log^p sin x dx</a>, Math. Comp. 40 (162) (1983) 565-570, r_{3,0}
%F A196878 Equals A019675*(6*A002117 + A002388*A002162 + 4*A002162^3).
%e A196878 6.041882909775093522150424130675995...
%p A196878 Pi/8*(6*Zeta(3)+Pi^2*log(2)+4*log(2)^3) ; evalf(%) ; # _R. J. Mathar_, Oct 08 2011
%t A196878 RealDigits[N[Pi/8 (6 Zeta[3] + Pi^2 Log[2] + 4 Log[2]^3), 150]][[1]]
%t A196878 Sqrt[Pi]/2*Derivative[3][Gamma[(#+1)/2]/Gamma[#/2+1]&][0] // RealDigits[#, 10, 105]& // First (* _Jean-François Alcover_, Mar 25 2013 *)
%o A196878 (PARI) Pi/8*(6*zeta(3)+Pi^2*log(2)+4*log(2)^3) \\ _G. C. Greubel_, Feb 12 2017
%Y A196878 Cf. A173623, A196877.
%K A196878 cons,nonn
%O A196878 1,1
%A A196878 _Seiichi Kirikami_, Oct 07 2011