login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A193716 Decimal expansion of Pi^3*log(2)/24 - 3*Pi*zeta(3)/16. 3
1, 8, 7, 4, 2, 6, 4, 2, 2, 8, 2, 8, 2, 3, 1, 0, 8, 0, 2, 6, 4, 5, 6, 9, 3, 1, 2, 2, 7, 3, 2, 7, 5, 0, 8, 1, 2, 5, 3, 0, 6, 9, 0, 1, 1, 7, 7, 0, 3, 1, 1, 5, 5, 7, 0, 8, 1, 0, 3, 2, 6, 0, 8, 3, 8, 8, 1, 8, 0, 2, 3, 3, 3, 1, 0, 6, 2, 0, 2, 8, 4, 9, 7, 6, 4, 9, 9, 2, 3, 1, 0, 6, 0, 2, 4, 4, 5, 8, 8, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The absolute value of the integral {x=0..Pi/2} x^2*log(sin(x )) dx or (d^2/da^2 (integral {x=0..Pi/2} cos(ax)*log(sin(x )) dx)) at a=0. The absolute value of (sum {n=1..infinity} (limit { a -> 0} (d^2/da^2 (sin((a+2n)*Pi/2)/n/(a+2n)))))-(Pi/2)^3*log(2)/3. [Seiichi Kirikami and Peter J. C. Moses]
REFERENCES
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, series and Products, 1.441.2, 4th edition, log(sin(x))=-(sum {1..infinity} cos(2nx)/n)-log(2).
LINKS
R. E. Crandall, J. P. Buhler, On the evaluation of Euler sums, Exper. Math. 3 (4) (1994) 275 (discuss int_{0..1} x^n*cot(x) dx which is obtained by partial integration).
S. Koyama and N. Kurokawa, Euler’s integrals and multiple sine functions, Proc. Amer. Math. Soc. 133(2005), 1257-1265.
FORMULA
Equals A091925*A002162/24-3*A000796*A002117/16.
EXAMPLE
0.18742642282823108026...
MATHEMATICA
RealDigits[ N[Pi (2 Pi^2 Log[2] - 9 Zeta[3]) / 48, 105] ][[1]]
PROG
(PARI) Pi^3*log(2)/24 - 3*Pi*zeta(3)/16 \\ Michel Marcus, Oct 25 2017
CROSSREFS
Sequence in context: A131081 A326919 A158288 * A268046 A260060 A260800
KEYWORD
cons,nonn
AUTHOR
Seiichi Kirikami, Aug 03 2011
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 12 06:13 EDT 2024. Contains 371623 sequences. (Running on oeis4.)