A261069 Decimal expansion of J_5 = Integral_{0..Pi/2} x^5/sin(x) dx.

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, 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, 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

Offset

1,1

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

J. M. Borwein, I. J. Zucker, and J. Boersma, The evaluation of character Euler double sums, The Ramanujan Journal, Volume 15, Issue 3 (April 2008), pp. 377-405; alternative link. See p. 13.

Formula

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

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.

Example

2.634318290518755162210315961284055055940934358931555842123212369587...

Mathematica

J5 = (5*Catalan*Pi^4)/8 - (29*I*Pi^6)/2016 - 30*I*Pi^2*PolyLog[4, -I] +
     240*I*PolyLog[6, -I]; RealDigits[J5 // Re, 10, 103] // First
(* Alternative: *)
RealDigits[NIntegrate[x^5/Sin[x], {x, 0, Pi/2}, WorkingPrecision->120]][[1]] (* Harvey P. Dale, Aug 09 2023 *)

Prog

(PARI) real(5*Catalan*Pi^4/8 - 29*I*Pi^6/2016 - 30*I*Pi^2*polylog(4, -I) + 240*I*polylog(6, -I)) \\ Amiram Eldar, Jun 27 2026

Crossrefs

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

Sequence in context: A066098 A299160 A123733 * A388368 A236557 A359243

Adjacent sequences: A261066 A261067 A261068 * A261070 A261071 A261072

Keyword

nonn,cons,changed

Author

Jean-François Alcover, Aug 08 2015

Status

approved