|
| |
|
|
A074962
|
|
Decimal expansion of Glaisher-Kinkelin constant A.
|
|
364
|
|
|
|
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
|
Arises in 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.
Named after the English mathematician and astronomer James Whitbread Lee Glaisher (1848-1928) and the Swiss mathematician Hermann Kinkelin (1832-1913). - Amiram Eldar, Jun 15 2021
|
|
|
REFERENCES
|
Steven R. Finch, Mathematical constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, p. 135.
Konrad Knopp, Theory and applications of infinite series, Dover, p. 555.
|
|
|
LINKS
|
Gheorghe Coserea, Table of n, a(n) for n = 1..10010
Chao-Ping Chen and Long Lin, Asymptotic expansions related to Glaisher-Kinkelin constant based on the Bell polynomials, Journal of Number Theory, Vol. 133 (2013), pp. 2699-2705.
Ovidiu Furdui, proposer, Problem 11494, Amer. Math. Monthly, Vol. 118, No. 9 (2011), 850-852.
J. W. L. Glaisher, On the Product 1^1.2^2.3^3...n^n, The Messenger of Mathematics, Vol. 7 (1878), pp. 43-47.
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J., Vol. 16 (2008), pp. 247-270; see Examples 5.2, 5.7, 5.11.
Fredrik Johansson et al., mpmath, Mathematical constants (Mpmath).
Fredrik Johansson et al., mpmath, Glaisher's constant to 20,000 digits.
Hermann Kinkelin, Über eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechnung, Journal für die reine und angewandte Mathematik, Vol. 57 (1860), pp. 122-138.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., Vol. 332, No. 1 (2007), pp. 292-314; see Section 5.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.
Wikipedia, Glaisher-Kinkelin constant.
|
|
|
FORMULA
|
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)).
|
|
|
EXAMPLE
|
1.2824271291006226368753425688697917277676889273250011920637400217404...
|
|
|
MAPLE
|
evalf(limit(product(k^k, k=1..n)/(n^(n^2/2+n/2+1/12)*exp(-n^2/4)), n=infinity), 120); # Vaclav Kotesovec, Oct 23 2014
|
|
|
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
|
Cf. A243262, A243263, A243264, A243265.
Cf. A000178, A002109, A051675, A255321, A255323, A255344.
Sequence in context: A257579 A199715 A296049 * A064863 A021358 A332353
Adjacent sequences: A074959 A074960 A074961 * A074963 A074964 A074965
|
|
|
KEYWORD
|
nonn,cons,nice
|
|
|
AUTHOR
|
Benoit Cloitre, Oct 05 2002
|
|
|
EXTENSIONS
|
More terms from Sascha Kurz, Feb 03 2003
|
|
|
STATUS
|
approved
|
| |
|
|
