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%I #69 Jun 16 2023 12:32:02
%S 0,7,2,8,1,5,8,4,5,4,8,3,6,7,6,7,2,4,8,6,0,5,8,6,3,7,5,8,7,4,9,0,1,3,
%T 1,9,1,3,7,7,3,6,3,3,8,3,3,4,3,3,7,9,5,2,5,9,9,0,0,6,5,5,9,7,4,1,4,0,
%U 1,4,3,3,5,7,1,5,1,1,4,8,4,8,7,8,0,8,6,9,2,8,2,4,4,8,4,4,0,1,4,6,0,4
%N Decimal expansion of the 1st Stieltjes constant gamma_1 (negated).
%C The Stieltjes constants are named after the Dutch mathematician Thomas Joannes Stieltjes (1856-1894). - _Amiram Eldar_, Jun 16 2021
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 166.
%H G. C. Greubel, <a href="/A082633/b082633.txt">Table of n, a(n) for n = 0..2500</a>
%H Dorin Andrica and Eugen J. Ionascu, <a href="http://www.emis.de/journals/ASUO/mathematics_/vol22-1/Andrica_D__Ionascu_E.J._nou-1__final_.pdf">On the number of polynomials with coefficients in [n]</a>, An. St. Univ. Ovidius Constanta, Volume XXII (2014), fascicola 1.
%H G. H. Hardy, <a href="https://babel.hathitrust.org/cgi/pt?id=inu.30000050138266&view=1up&seq=225">Note on Dr. Vacca's series for gamma</a>, Quart. J. Pure Appl. Math., Vol. 43 (1912), pp. 215-216. [Available only in the USA]
%H Krzysztof Maślanka and Andrzej Koleżyński, <a href="https://arxiv.org/abs/2210.04609">The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm</a>, arXiv preprint, arXiv:2210.04609 [math.NT], 2022.
%H Marc Prévost, <a href="https://doi.org/10.48550/arXiv.2305.15806">Expansion of generalized Stieltjes constants in terms of derivatives of Hurwitz zeta-functions</a>, arXiv:2305.15806 [math.NA], 2023.
%H Sandeep Tyagi, <a href="https://arxiv.org/abs/2212.07956">High precision computation and a new asymptotic formula for the generalized Stieltjes constants</a>, arXiv preprint, arXiv:2212.07956 [math.NA], 2022.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StieltjesConstants.html">Stieltjes Constants</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Stieltjes_constants">Stieltjes constants</a>.
%F Equals lim_{y->infinity} y*(Im(zeta(1+i/y))+y).
%F Equals lim_{n->infinity} (((log(n))^2)/2 - Sum_{k=2..n} (log(k))/k). - _Warut Roonguthai_, Aug 04 2005
%F Equals Integral_{0..infinity} (coth(Pi*x)-1)*(x*log(1+x^2)-2*arctan(x))/(2*(1+x^2)) dx. - _Jean-François Alcover_, Jan 28 2015
%F Using the abbreviations a = log(z^2 + 1/4)/2, b = arctan(2*z) and c = cosh(Pi*z) then gamma_1 = -(Pi/2)*Integral_{0..infinity} (a^2 - b^2)/c^2. The general case is for n >= 0 (which includes Euler's gamma as gamma_0) gamma_n = -(Pi/(n+1))* Integral_{0..infinity} sigma(n+1)/c^2, where sigma(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*b^(2*k)*a^(n-2*k). - _Peter Luschny_, Apr 19 2018
%F Equals log(2)^2/6 - log(2)*gamma/2 + (1/(2*log(2)) * Sum_{k>=1} (-1)^k * log(k)^2/k, where gamma is Euler's constant (A001620) (Hardy, 1912). - _Amiram Eldar_, Jun 09 2023
%F Equals Sum_{j>=1} Zeta'(2*j + 1) / (2*j + 1). - _Peter Luschny_, Jun 16 2023
%e -0.0728158454836767248605863758749...
%p evalf(gamma(1)) ; # _R. J. Mathar_, Sep 15 2013
%t Prepend[RealDigits[c=N[StieltjesGamma[1], 120], 10][[1]], 0]
%t N[EulerGamma^2 - Residue[Zeta[s]^3, {s, 1}]/3, 100] (* _Vaclav Kotesovec_, Jan 07 2017 *)
%o (PARI) intnum(x=0,oo,(1/tanh(Pi*x)-1)*(x*log(1+x^2)-2*atan(x))/(2*(1+x^2))) \\ _Charles R Greathouse IV_, Mar 10 2016
%o (PARI) Stieltjes(n)=my(a=log(2)); a^n/(n+1)*sumalt(k=1,(-1)^k/k*subst(bernpol(n+1),'x,log(k)/a))
%o Stieltjes(1) \\ _Charles R Greathouse IV_, Feb 23 2022
%Y Cf. A001620, A086279, A086280, A086281, A086282, A183141, A183167, A183206, A184853, A184854.
%K cons,nonn
%O 0,2
%A _Benoit Cloitre_, May 24 2003
%E More terms from _Eric W. Weisstein_, Jul 14 2003
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