cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

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

Views

Author

Seiichi Kirikami, Oct 07 2011

Keywords

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

Crossrefs

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).

A318741 Decimal expansion of Pi^3/48 + Pi*log(2)^2/4.

Original entry on oeis.org

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

Views

Author

Hugo Pfoertner, Sep 17 2018

Keywords

Comments

The first part of Ramanujan's question 308 in the Journal of the Indian Mathematical Society (III, 168) asked "Show that Integral_{t=0..Pi/2} t * cotan(t) * log(sin(t)) dt = -Pi^3/48 - Pi*log(2)^2/4".

Examples

			1.0233110122363703230848205040884867383187209767473281303051342763...
		

Programs

  • Mathematica
    RealDigits[Pi^3/48 + Pi*Log[2]^2/4, 10, 100][[1]] (* Amiram Eldar, Oct 04 2021 *)
  • PARI
    Pi^3/48+Pi*log(2)^2/4
    
  • PARI
    -intnum(x=0,Pi/2,x*cotan(x)*log(sin(x)))

Formula

Equals Sum_{k>=0} binomial(2*k,k)/(4^k*(2*k+1)^3) (Ribeiro, 2018). - Amiram Eldar, Oct 04 2021
Equals 4F3(1/2,1/2,1/2,1/2 ; 3/2,3/2,3/2 ; 1) [Adamchik]. - R. J. Mathar, Aug 19 2024
Equals A196877/2. - R. J. Mathar, Aug 23 2024
Showing 1-2 of 2 results.