A084448 Decimal expansion of (negative of) Kinkelin constant.

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

Offset

0,2

Comments

Named after the Swiss mathematician Hermann Kinkelin (1832-1913). - Amiram Eldar, Jun 16 2021

Links

Table of n, a(n) for n=0..98.

Gert Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., Vol. 7, No. 4 (1998), pp. 343-359.

Hermann Kinkelin, Ueber eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechnung, J. Reine Angew. Math., Vol. 57 (1860), pp. 122-158; alternative link. See eq. (22), p. 133.

E. M. Wright, Asymptotic partition formulae, I: Plane partitions, Quart. J. Math., Vol. 2 (1931), pp. 177-189.

Formula

Zeta(1, -1). Almkvist gives many formulas.

Equals (1 - gamma - log(2*Pi))/12 + Zeta'(2)/(2*Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 25 2015

From Amiram Eldar, Jun 16 2021: (Start)

Equals 1/24 - gamma/3 - Sum_{k>=1} (zeta(2*k+1)-1)/((2*k+1)*(2*k+3)) = 1/12 - log(A), where A is the Glaisher-Kinkelin constant (A074962) (Kinkelin, 1860).

Equals 2 * Integral_{x>=0} x*log(x)/(exp(2*Pi*x)-1) dx (Wright, 1931). (End)

Example

-0.1654211437004509292139196602427806427640363803352017836665223...

Maple

Digits := 200; evalf(Zeta(1, -1));

Mathematica

RealDigits[1/12 - Log[Glaisher], 10, 99] // First (* Jean-François Alcover, Feb 15 2013 *)

Prog

(PARI) -zeta'(-1) \\ Charles R Greathouse IV, Dec 12 2013

Crossrefs

Cf. A001620, A074962, A084539.

Sequence in context: A109063 A110390 A193178 * A256596 A173431 A263879

Adjacent sequences: A084445 A084446 A084447 * A084449 A084450 A084451

Keyword

nonn,cons,easy

Author

N. J. A. Sloane, Jun 27 2003

Status

approved