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A074962 Decimal expansion of Glaisher-Kinkelin constant A. 27
1, 2, 8, 2, 4, 2, 7, 1, 2, 9, 1, 0, 0, 6, 2, 2, 6, 3, 6, 8, 7, 5, 3, 4, 2, 5, 6, 8, 8, 6, 9, 7, 9, 1, 7, 2, 7, 7, 6, 7, 6, 8, 8, 9, 2, 7, 3, 2, 5, 0, 0, 1, 1, 9, 2, 0, 6, 3, 7, 4, 0, 0, 2, 1, 7, 4, 0, 4, 0, 6, 3, 0, 8, 8, 5, 8, 8, 2, 6, 4, 6, 1, 1, 2, 9, 7, 3, 6, 4, 9, 1, 9, 5, 8, 2, 0, 2, 3, 7, 4, 3, 9, 4, 2, 0, 6, 4, 6, 1, 2, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Arise in various asymptotic expressions such as A002109(n)=1^1*2^2*3^3*...*n^n which is asymptotic to A*n^(n^2/2+n/2+1/12)*exp(-n^2/4). See A002109 for more references and links.

REFERENCES

S. R. Finch, Mathematical constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 135.

K. Knopp, "Theory and applications of infinite series", Dover, p. 555.

LINKS

Table of n, a(n) for n=1..111.

Chao-Ping Chen and Long Lin, Asymptotic expansions related to Glaisher-Kinkelin constant based on the Bell polynomials, Journal of Number Theory 133 (2013) 2699-2705.

O. Furdui, proposer, Problem 11494, Amer. Math. Monthly, 118 (2011), 850-852.

J. W. L. Glaisher, On the Product 1^1.2^2.3^3...n^n, The Messenger of Mathematics 7 (1878), 43-47.

J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J., 16 (2008), 247-270; see Examples 5.2, 5.7, 5.11.

H. Kinkelin, Über eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechnung, Journal für die reine und angewandte Mathematik 57 (1860), 122-138.

Fredrik Johansson et al.,  mpmath, Glaisher's constant to 20,000 digits.

J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., 332 (2007), 292-314; see Section 5.

Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant

FORMULA

A=1.2824271291... A = 2^(1/36)*Pi^(1/6)*exp(1/3*(-Gamma/4+s(2)/3-s(3)/4+...)) where s(k) denotes sum(n>=0, 1/(2n+1)^k). Closed expressions for A are exp(-zeta'(2)/2/Pi^2 + log(2*Pi)/12 + Gamma/12) or exp(1/12-zeta'(-1)).

MATHEMATICA

RealDigits[ Glaisher, 10, 111][[1]] (* Robert G. Wilson v *)

PROG

(PARI) x=10^(-100); exp(1/12-(zeta(-1+x)-zeta(-1))/x)

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

CROSSREFS

Sequence in context: A173686 A090975 A199715 * A064863 A021358 A203022

Adjacent sequences:  A074959 A074960 A074961 * A074963 A074964 A074965

KEYWORD

cons,nonn

AUTHOR

Benoit Cloitre, Oct 05 2002

EXTENSIONS

More terms from Sascha Kurz, Feb 03 2003

STATUS

approved

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Last modified September 18 22:41 EDT 2014. Contains 246937 sequences.