A193717 -id:A193717 - cp's OEIS frontend

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

Seiichi Kirikami, Aug 03 2011

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]

			0.18742642282823108026...

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).

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.

Cf. A173623, A173624, A193717.

RealDigits[ N[Pi (2 Pi^2 Log[2] - 9 Zeta[3]) / 48, 105] ][[1]]

Pi^3*log(2)/24 - 3*Pi*zeta(3)/16 \\ Michel Marcus, Oct 25 2017

Equals A091925*A002162/24-3*A000796*A002117/16.

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

Original entry on oeis.org

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

Seiichi Kirikami, Sep 01 2011

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

			0.12204729588592872163...

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

Cf. A173623, A173624, A193716, A193717.

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

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

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

Original entry on oeis.org

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

Seiichi Kirikami, Sep 01 2011

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).

			0.11757583407233248206...

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

Cf. A173623, A173624, A193716, A193717, A196456.

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]]

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

cp's OEIS Frontend

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

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

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

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cp's OEIS Frontend

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

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

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

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

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

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