A243262 Decimal expansion of the generalized Glaisher-Kinkelin constant A(2).

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

Offset

1,3

Comments

Also known as the second Bendersky constant.

This is likely the same as the constant B considered in section 3 of the Choi and Srivastava link. - R. J. Mathar, Oct 03 2016

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.

Links

G. C. Greubel, Table of n, a(n) for n = 1..2002

J. Choi and H. M. Srivastava, Certain classes of series involving the zeta function, J. Math. Annal. Applic. 231 (1999) 91-117.

K. Kimoto, N. Kurokawa, C. Sonoki, M. Wakayama, Some examples of generalized zeta regularized products, Kodai Math. J. 27 (2004), 321-335.

Tobias Kyrion, A closed-form expression for zeta(3), arXiv:2008.05573 [math.GM], 2020.

Eric Weisstein's MathWorld, Glaisher-Kinkelin Constant

Formula

A(k) = exp(B(k+1)/(k+1)*H(k)-zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.

A(0) = sqrt(2*Pi) (A019727),

A(1) = A = Glaisher-Kinkelin constant (A074962),

A(2) = exp(-zeta'(-2)) = exp(zeta(3)/(4*Pi^2)).

Equals exp(-A240966). - Vaclav Kotesovec, Feb 22 2015

Example

1.03091675219739211419331309646694229...

Mathematica

RealDigits[Exp[Zeta[3]/(4*Pi^2)], 10, 99] // First
RealDigits[Exp[N[(BernoulliB[2]/4)*(Zeta[3]/Zeta[2]), 200]]]//First (* G. C. Greubel, Dec 31 2015 *)

Prog

(PARI) exp(zeta(3)/(4*Pi^2)) \\ Felix Fröhlich, Jun 27 2019

Crossrefs

Cf. A051675, A055462, A240966, A255269.

Cf. A019727, A074962, A243263, A243264, A243265, A266553, A266554, A266555, A266556, A266557, A266558, A266559, A260662, A266560, A266562, A266563, A266564, A266565, A266566, A266567.

Cf. A002117 (zeta(3)).

Sequence in context: A096429 A196622 A196827 * A191661 A296487 A159760

Adjacent sequences: A243259 A243260 A243261 * A243263 A243264 A243265

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 02 2014

Status

approved