A173624 Decimal expansion of Pi^2*log(2)/8 - 7*zeta(3)/16, where zeta is the Riemann zeta function.

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

Offset

0,1

Links

Table of n, a(n) for n=0..104.

Chenli Li, Wenchang Chu, Improper integrals involving powers of Inverse Trigonometric and hyperbolic Functions, Mathematics 10 (16) (2022) 2980

Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (7.3.1)

Kazuhiro Onodera, Generalized log sine integrals and the Mordell-Tornheim zeta values, Trans. Am. Math. Soc. 363 (3) (2010), 1463-1485.

Formula

The absolute value of the Integral_{x=0..Pi/2} x*log(sin(x)) dx.

Equals A111003 * A002162 - 7*A002117/16. [typo corrected by R. J. Mathar, Nov 15 2010]

Equals Sum_{n>=1} (phi(-1,1,2n)/(2n-1)^2), where phi is the Lerch transcendent. - John Molokach, Jul 22 2013

Equals Sum_{n>=1} (4^n / (8n^3 binomial(2n,n)). - John Molokach, Aug 01 2013

Equals Integral_{y=0..1} Integral_{x=0..1} log(x*y+1)/(1-(x*y)^2) dx dy. - Amiram Eldar, Apr 17 2022

Example

0.3292361628498170682435494485830026...

Maple

-7*Zeta(3)/16+Pi^2*log(2)/8 ; evalf(%) ;

Mathematica

N[(1/8) (Pi^2 Log[2] - 7 Zeta[3]/2), 100] (* John Molokach, Aug 02 2013 *)

Crossrefs

Cf. A002117, A002162, A046161, A111003.

Sequence in context: A010271 A291777 A143074 * A182023 A349213 A228936

Adjacent sequences: A173621 A173622 A173623 * A173625 A173626 A173627

Keyword

cons,nonn

Author

R. J. Mathar, Nov 08 2010

Status

approved