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 A074962 Decimal expansion of Glaisher-Kinkelin constant A. 441
 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. Robert A. Van Gorder, Glaisher-type products over the primes, International Journal of Number Theory, Vol. 8, No. 2 (2012), pp. 543-550. 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)). Equals (2*Pi)^(1/4) / limit_{n->oo} Product_{k=1..n} Gamma(k/n)^(k/n^2). - Vaclav Kotesovec, Dec 02 2023 Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^4-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(2)/2 = 1/12 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024 Equals e^(-1/4 + Integral_{x=1..2} x*log(sqrt(2*Pi)) - B_2(x) + x^2*Psi(x)/2 dx), where B_2(x) is the second Bernoulli polynomial and Psi(x) is the digamma function. - Andrea Pinos, Apr 16 2024 Equals Product_{k>=1} 2^(10^(-k) + 3/13^k)((2*k)/(2*k + 1))^((k/3 + 1/12))((2*k + 2)/(2*k + 1))^((k/3 + 1/4)). - Antonio Graciá Llorente, May 20 2024 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. A001620, 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

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