This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A002389 M3740 #40 Aug 28 2025 00:33:06
%S A002389 5,4,9,5,3,9,3,1,2,9,8,1,6,4,4,8,2,2,3,3,7,6,6,1,7,6,8,8,0,2,9,0,7,7,
%T A002389 8,8,3,3,0,6,9,8,9,8,1,2,6,3,0,6,4,7,9,1,0,9,0,1,5,1,3,0,4,5,7,6,6,3,
%U A002389 1,4,2,0,0,5,5,7,5,3,0,4,7,5,6,2,6,1,8
%N A002389 Decimal expansion of -log(gamma), where gamma is Euler's constant A001620.
%C A002389 From _Peter Bala_, Aug 24 2025: (Start)
%C A002389 By definition, the Euler-Mascheroni constant gamma = lim_{n -> oo} s(n), where s(n) = Sum_{k = 1..n} 1/k - log(n). The convergence is slow. For example, s(50) = 0.5(87...) is only correct to 1 decimal digit. Let S(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*s(n+k). Elsner shows that S(n) converges to gamma much more rapidly. For example, S(50) = 0.57721566490153286060651209008(02...) gives gamma correct to 29 decimal digits.
%C A002389 Define E(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*log(s(n+k)). Then it appears that E(n) converges rapidly to log(gamma). For example, E(50) = -0.549539312981644822337661768802(88...) gives log(gamma) correct to 30 decimal digits. Cf. A073004. (End)
%D A002389 W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
%D A002389 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002389 Ivan Panchenko, <a href="/A002389/b002389.txt">Table of n, a(n) for n = 0..1000</a>
%H A002389 C. Elsner, <a href="https://doi.org/10.1090/S0002-9939-1995-1233969-4">On a sequence transformation with integral coefficients for Euler's constant</a>, Proc. Amer. Math. Soc., Vol. 123 (1995), Number 5, pp. 1537-1541.
%H A002389 Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/loggamma.txt">-log(gamma) to 10000 digits</a>
%H A002389 Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap63.html">-log(gamma) to 1024 digits</a>
%e A002389 .549539312981644822337661768802907788330698981263...
%t A002389 RealDigits[-Log[EulerGamma], 10, 100][[1]] (* _G. C. Greubel_, Sep 07 2018 *)
%o A002389 (PARI) -log(Euler) \\ _Michel Marcus_, Mar 11 2013
%o A002389 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); -Log(EulerGamma(R)); // _G. C. Greubel_, Sep 07 2018
%Y A002389 Cf. A001620, A073004, A155969, A213440.
%K A002389 nonn,cons,changed
%O A002389 0,1
%A A002389 _N. J. A. Sloane_