A196878 Decimal expansion of (Pi/8)*(6*zeta(3)+Pi^2*log(2)+4*log(2)^3).

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

Offset: 1

Author

Seiichi Kirikami, Oct 07 2011

Keywords

• cons
• nonn

Comments

The absolute value of the integral {x=0..Pi/2} log(sin(x))^3 dx. The absolute value of m=3 of sqrt(Pi)/2*(d^m/da^m(gamma((a+1)/2)/gamma(a/2+1))) at a=0. - Seiichi Kirikami and Peter J. C. Moses, Oct 07 2011

Examples

			6.041882909775093522150424130675995...
		

References

• I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 3.621.1

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• K. S. Kolbig, On the integral int_0^Pi/2 log^n cos x log^p sin x dx, Math. Comp. 40 (162) (1983) 565-570, r_{3,0}

Crossrefs

Cf. A173623, A196877.

Programs

Maple

Pi/8*(6*Zeta(3)+Pi^2*log(2)+4*log(2)^3) ; evalf(%) ; # R. J. Mathar, Oct 08 2011

Mathematica

RealDigits[N[Pi/8 (6 Zeta[3] + Pi^2 Log[2] + 4 Log[2]^3), 150]][[1]]
Sqrt[Pi]/2*Derivative[3][Gamma[(#+1)/2]/Gamma[#/2+1]&][0] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 25 2013 *)

PARI

Pi/8*(6*zeta(3)+Pi^2*log(2)+4*log(2)^3) \\ G. C. Greubel, Feb 12 2017

Formula

Equals A019675*(6*A002117 + A002388*A002162 + 4*A002162^3).