A194657 Decimal expansion of (4*Pi^6*log(2) - 90*Pi^4*zeta(3) + 1350*Pi^2*zeta(5) - 5715*zeta(7))/1536.

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

Offset: 0

Author

Seiichi Kirikami, Sep 01 2011

Keywords

• cons
• nonn

Comments

The absolute value of the integral {x=0..Pi/2} x^5*log(sin(x )) dx or (d^5/da^5 (integral {x=0..Pi/2} sin(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+1)/da^(2m+1) ((1-cos((a+2n)*Pi/2))/n/(a+2n)))))-(pi/2)^2(m+1)*log(2)/2/(m+1).

Examples

			0.11757583407233248206...
		

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, A196456.

Programs

Mathematica

RealDigits[ N[(4 Pi^6*Log[2]-90 Pi^4*Zeta[3]+1350 Pi^2*Zeta[5]-5715 Pi^2*Zeta[7])/1536,150]][[1]]

Formula

Equals (4*A092732*A002162-90*A092425*A002117+1350*A002388*A013663-5715*A013665)/1536.