A243264 Decimal expansion of the generalized Glaisher-Kinkelin constant A(4).

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

Offset: 0

Author

Jean-François Alcover, Jun 02 2014

Keywords

• nonn
• cons

Comments

Also known as the 4th Bendersky constant.

Examples

			0.9920479745250402600134369776254433567369...
		

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 = 0..2002
• Eric Weisstein's MathWorld, Glaisher-Kinkelin Constant.

Crossrefs

Cf. A255323, A259069.

Cf. A019727, A074962, A243262, A243263, A243265, A266553, A266554, A266555, A266556, A266557, A266558, A266559, A260662, A266560, A266562, A266563, A266564, A266565, A266566, A266567.

Programs

Mathematica

RealDigits[Exp[-3*Zeta[5]/(4*Pi^4)], 10, 98] // First
RealDigits[Exp[N[(BernoulliB[4]/4)*(Zeta[5]/Zeta[4]), 100]]] // First (* G. C. Greubel, Dec 31 2015 *)

PARI

exp(-3*zeta(5)/(4*Pi^4)) \\ Stefano Spezia, Dec 01 2024

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(4) = exp(-zeta'(-4)) = exp(-3*zeta(5)/(4*Pi^4)).

A(4) = exp((B(4)/4)*(zeta(5)/zeta(4))). - G. C. Greubel, Dec 31 2015