A193716 - OEIS

%I #25 Oct 25 2017 05:10:18

%S 1,8,7,4,2,6,4,2,2,8,2,8,2,3,1,0,8,0,2,6,4,5,6,9,3,1,2,2,7,3,2,7,5,0,

%T 8,1,2,5,3,0,6,9,0,1,1,7,7,0,3,1,1,5,5,7,0,8,1,0,3,2,6,0,8,3,8,8,1,8,

%U 0,2,3,3,3,1,0,6,2,0,2,8,4,9,7,6,4,9,9,2,3,1,0,6,0,2,4,4,5,8,8,1

%N Decimal expansion of Pi^3*log(2)/24 - 3*Pi*zeta(3)/16.

%C The absolute value of the integral {x=0..Pi/2} x^2*log(sin(x )) dx or (d^2/da^2 (integral {x=0..Pi/2} cos(ax)*log(sin(x )) dx)) at a=0. The absolute value of (sum {n=1..infinity} (limit { a -> 0} (d^2/da^2 (sin((a+2n)*Pi/2)/n/(a+2n)))))-(Pi/2)^3*log(2)/3. [_Seiichi Kirikami_ and _Peter J. C. Moses_]

%D I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, series and Products, 1.441.2, 4th edition, log(sin(x))=-(sum {1..infinity} cos(2nx)/n)-log(2).

%H R. E. Crandall, J. P. Buhler, <a href="https://projecteuclid.org/euclid.em/1048515810">On the evaluation of Euler sums</a>, Exper. Math. 3 (4) (1994) 275 (discuss int_{0..1} x^n*cot(x) dx which is obtained by partial integration).

%H S. Koyama and N. Kurokawa, <a href="https://doi.org/10.1090/S0002-9939-04-07863-3">Euler’s integrals and multiple sine functions</a>, Proc. Amer. Math. Soc. 133(2005), 1257-1265.

%F Equals A091925*A002162/24-3*A000796*A002117/16.

%e 0.18742642282823108026...

%t RealDigits[ N[Pi (2 Pi^2 Log[2] - 9 Zeta[3]) / 48, 105] ][[1]]

%o (PARI) Pi^3*log(2)/24 - 3*Pi*zeta(3)/16 \\ _Michel Marcus_, Oct 25 2017

%Y Cf. A173623, A173624, A193717.

%K cons,nonn

%O 0,2

%A _Seiichi Kirikami_, Aug 03 2011