A396329 Decimal expansion of lim_{s->1} Pi^(s/2)/(s*(s-1)*Gamma(s/2)) - 1/(s-1).

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

Offset

0,1

References

I. S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products, 7th ed., Academic Press, 2007, p. 658, eq. (6.468).

Links

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

Junesang Choi and H. M. Srivastava, Integral Representations for the Euler-Mascheroni Constant gamma, Integral Transforms and Special Functions, Vol. 21, No. 9 (2010), pp. 675-690.

Donal F. Connon, Some trigonometric integrals involving the log gamma and the digamma function, arXiv:1005.3469 [math.CA], 2010, p. 126, eq. (5.9).

William FitzGerald, Exactly solvable interacting particle systems, PhD thesis, University of Warwick, 2019. See p. 151, eq. (3.63).

Will FitzGerald, Roger Tribe, and Oleg Zaboronski, Sharp asymptotics for Fredholm Pfaffians related to interacting particle systems and random matrices, Electron. J. Probab., Vol. 25 (2020), paper 116. See p. 8, eq. (2.28).

Niels Nielsen, Handbuch der theorie der gammafunktion, Teubner, Leipzig, 1906, see p. 204, eq. (11).

Formula

Equals log(2*sqrt(Pi)) - 1 + gamma/2, where gamma is Euler's constant (A001620).

Equals gamma - Integral_{t>=1} (1 + sqrt(t))*(omega(t)/t) dt, where omega(t) = Sum_{n>=1} exp(-n^2*Pi*t).

Equals -(1 + Integral_{x=0..1} psi(x) * sin(2*Pi*x)^2 dx), where psi the digamma function.

Example

0.554119955935411826792201842175907139420227208787872...

Mathematica

RealDigits[Log[2*Sqrt[Pi]] - 1 + EulerGamma/2, 10, 120][[1]]

Prog

(PARI) log(2) + log(Pi)/2 - 1 + Euler/2

Crossrefs

Cf. A001620, A155968.

Sequence in context: A202695 A110986 A193721 * A377202 A386459 A019927

Adjacent sequences: A396326 A396327 A396328 * A396330 A396331 A396332

Keyword

nonn,cons

Author

Amiram Eldar, May 22 2026

Status

approved