A225125 -id:A225125 - cp's OEIS frontend

A261068 Decimal expansion of J_4 = Integral_{0..Pi/2} x^4/sin(x) dx.

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

Offset: 1

Jean-François Alcover, Aug 08 2015

			2.05316073148059166895654129602651136685655884457239569438518892765...

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, April 2008, Volume 15, Issue 3, pp 377-405, see p. 13.

Cf. A006752 (J_1 / 2 = Catalan's constant), A245073 (J_2), A225125 (J_3), A261069 (J_5).

J4 = Catalan*Pi^3 - 7*I*Pi^5/480 - 24*I*Pi*PolyLog[4, -I] + 93*Zeta[5]/2; RealDigits[J4 // Re, 10, 102] // First

J_4 = Catalan*Pi^3 - 7*i*Pi^5/480 - 24*i*Pi*PolyLog(4, -i) + (93*zeta(5))/2.

Also equals Catalan*Pi^3 + (1/64)*(Pi*(PolyGamma(3, 3/4) - PolyGamma(3, 1/4)) + 2976*Zeta(5));

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

Original entry on oeis.org

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

Jean-François Alcover, Aug 08 2015

			2.634318290518755162210315961284055055940934358931555842123212369587...

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, April 2008, Volume 15, Issue 3, pp 377-405, see p. 13.

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

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

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

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.

cp's OEIS Frontend

A261068 Decimal expansion of J_4 = Integral_{0..Pi/2} x^4/sin(x) dx.

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