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 A074962 Decimal expansion of Glaisher-Kinkelin constant A. 292
 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 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 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 = 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 A203022 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

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Last modified August 19 23:06 EDT 2018. Contains 313900 sequences. (Running on oeis4.)