A193716 Decimal expansion of Pi^3*log(2)/24 - 3*Pi*zeta(3)/16.

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

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

Table of n, a(n) for n=0..99.

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

Cf. A173623, A173624, A193717.

Sequence in context: A131081 A326919 A158288 * A268046 A260060 A260800

Adjacent sequences: A193713 A193714 A193715 * A193717 A193718 A193719

Keyword

cons,nonn

Author

Seiichi Kirikami, Aug 03 2011

Status

approved