A261069 - OEIS

%I #15 Aug 09 2023 12:08:24

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

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

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

%N Decimal expansion of J_5 = Integral_{0..Pi/2} x^5/sin(x) dx.

%H G. C. Greubel, <a href="/A261069/b261069.txt">Table of n, a(n) for n = 1..1000</a>

%H J. M. Borwein, I. J. Zucker and J. Boersma, <a href="http://carma.newcastle.edu.au/MZVs/mzv-week05.pdf">The evaluation of character Euler double sums</a>, The Ramanujan Journal, April 2008, Volume 15, Issue 3, pp 377-405, see p. 13.

%F J_5 = (5*Catalan*Pi^4)/8 - (29*i*Pi^6)/2016 - 30*i*Pi^2*PolyLog(4, -i) + 240*i*PolyLog(6, -i).

%F Also equals (40*Pi^2*(32*Catalan*Pi^2 - PolyGamma(3, 1/4) + PolyGamma(3, 3/4)) + PolyGamma(5, 1/4) - PolyGamma(5, 3/4))/2048.

%e 2.634318290518755162210315961284055055940934358931555842123212369587...

%t J5 = (5*Catalan*Pi^4)/8 - (29*I*Pi^6)/2016 - 30*I*Pi^2*PolyLog[4, -I] +

%t      240*I*PolyLog[6, -I]; RealDigits[J5 // Re, 10, 103] // First

%t RealDigits[NIntegrate[x^5/Sin[x],{x,0,Pi/2},WorkingPrecision->120]][[1]] (* _Harvey P. Dale_, Aug 09 2023 *)

%Y Cf. A006752 (J_1 / 2 = Catalan's constant), A245073 (J_2), A225125 (J_3), A261068 (J_4).

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Aug 08 2015