A194656 Decimal expansion of (2*Pi^5*log(2) - 30*Pi^3*zeta(3) + 225*Pi*zeta(5))/320.

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

Offset: 0

Author

Seiichi Kirikami, Sep 01 2011

Keywords

• cons
• nonn

Comments

The absolute value of the integral{x=0..Pi/2} x^4*log(sin(x )) dx or(d^4/da^4(integral {x=0..Pi/2} cos(ax)*log(sin(x )) dx)) at a=0. The absolute value of m=2 of (-1)^(m+1)*(sum {n=1..infinity} (limit {a -> 0} (d^(2m)/da^(2m)(sin((a+2n)*Pi/2)/n/(a+2n)))))-(Pi/2)^(2m+1)*log(2)/(2m+1). - Seiichi Kirikami and Peter J. C. Moses, Sep 01 2011

Examples

			0.12204729588592872163...
		

References

• I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 1.441.2

Crossrefs

Cf. A173623, A173624, A193716, A193717.

Programs

Mathematica

RealDigits[ N[Pi (2 Pi^4*Log[2]-30 Pi^2*Zeta[3]+225 Zeta[5])/320, 150]][[1]]

Formula

Equals (2*A092731*A002162-30*A091925*A002117+225*A000796*A013663)/320.