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%I #18 Feb 25 2013 11:46:06
%S 1,1,7,5,7,5,8,3,4,0,7,2,3,3,2,4,8,2,0,6,2,4,2,9,0,6,7,9,4,9,1,4,7,5,
%T 8,4,3,3,4,1,6,4,3,8,9,9,8,1,6,2,9,0,8,8,8,6,9,5,3,0,2,4,7,6,4,9,1,9,
%U 1,2,8,4,2,7,1,5,5,9,4,7,1,1,8,2,6,8,8,8,9,0,0,3,1,4,1,1,5,9,4,4,7,1,9,9,4
%N Decimal expansion of (4*Pi^6*log(2) - 90*Pi^4*zeta(3) + 1350*Pi^2*zeta(5) - 5715*zeta(7))/1536.
%C The absolute value of the integral {x=0..Pi/2} x^5*log(sin(x )) dx or (d^5/da^5 (integral {x=0..Pi/2} sin(ax)*log(sin(x )) dx)) at a=0. The absolute value of m=2 of (-1)^(m+1)*(sum {n=1..infinity} (limit {a -> 0} (d^(2m+1)/da^(2m+1) ((1-cos((a+2n)*Pi/2))/n/(a+2n)))))-(pi/2)^2(m+1)*log(2)/2/(m+1).
%D I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 1.441.2
%F Equals (4*A092732*A002162-90*A092425*A002117+1350*A002388*A013663-5715*A013665)/1536.
%e 0.11757583407233248206...
%t RealDigits[ N[(4 Pi^6*Log[2]-90 Pi^4*Zeta[3]+1350 Pi^2*Zeta[5]-5715 Pi^2*Zeta[7])/1536,150]][[1]]
%Y Cf. A173623, A173624, A193716, A193717, A196456.
%K cons,nonn
%O 0,3
%A _Seiichi Kirikami_, Sep 01 2011
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