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%I #30 Nov 30 2020 08:42:15
%S 3,2,5,9,5,5,7,5,1,5,9,1,7,9,1,0,1,9,5,2,5,0,8,7,4,5,8,2,6,7,6,5,5,9,
%T 2,5,7,9,7,6,4,7,2,2,0,4,3,9,9,4,3,0,0,4,8,1,1,7,9,7,4,8,6,7,3,8,9,7,
%U 9,3,7,0,1,4,9,5,4,6,8,7,9,2,4,7,8,9,6,5,2,5,8,8,2,0,0,8,6,7,3,8,0,4,3,2
%N Decimal expansion of gamma_1(1/3), the first generalized Stieltjes constant at 1/3 (negated).
%H G. C. Greubel, <a href="/A254331/b254331.txt">Table of n, a(n) for n = 1..5000</a>
%H Iaroslav V. Blagouchine, <a href="http://arxiv.org/abs/1401.3724">A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments</a>, arXiv:1401.3724 [math.NT], 2015.
%H Iaroslav V. Blagouchine, <a href="http://www.researchgate.net/publication/259743013_A_theorem_for_the_closed-form_evaluation_of_the_first_generalized_Stieltjes_constant_at_rational_arguments_and_some_related_summations">A theorem ... (same title)</a>, Journal of Number Theory Volume 148, March 2015, Pages 537-592.
%H Iaroslav V. Blagouchine, <a href="http://link.springer.com/article/10.1007/s11139-013-9528-5">Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results</a>, The Ramanujan Journal October 2014, Volume 35, Issue 1, pp 21-110.
%H Iaroslav V. Blagouchine, <a href="http://www.researchgate.net/publication/257381156_Rediscovery_of_Malmsten%27s_integrals_their_evaluation_by_contour_integration_methods_and_some_related_results">Rediscovery of Malmsten’s integrals: Full PDF text</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>.
%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 integral_[0..infinity] (3*(-2*arctan(3*x) + 3*x*log(1/9 + x^2)))/((-1 + exp(2*Pi*x))*(1 + 9*x^2)) dx - 3*log(3)/2 - log(3)^2/2.
%e -3.25955751591791019525087458267655925797647220439943...
%p evalf(int((3*(-2*arctan(3*x)+3*x*log(1/9+x^2)))/((-1+exp(2*Pi*x))*(9*x^2+1)), x = 0..infinity)-3*log(3)*(1/2)-(1/2)*log(3)^2, 120); # _Vaclav Kotesovec_, Jan 29 2015
%t gamma1[1/3] = (1/6)*((-Sqrt[3])*Pi*(EulerGamma + Log[(24*Pi^(5/2))/Gamma[1/6]^3]) - 3*(Log[3]^2 + EulerGamma*Log[27] + 2*Log[3]*Log[2*Pi] + 2*Log[2*Pi]^2 + Log[3/(4*Pi^2)]*Log[6*Pi] - 2*StieltjesGamma[1] + Derivative[2, 0][Zeta][0, 1/3] + Derivative[2, 0][Zeta][0, 2/3])) // Re; RealDigits[gamma1[1/3], 10, 104] // First
%t (* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 1/3], 10, 104] // First
%Y Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254345 (gamma_1(2/3)), A254347 (gamma_1(1/4)), A254348 (gamma_1(3/4)), A254349 (gamma_1(1/6)), A254350 (gamma_1(5/6)), A251866 (gamma_1(1/5)), A255188 (gamma_1(1/8)), A255189 (gamma_1(1/12)).
%K nonn,cons,easy
%O 1,1
%A _Jean-François Alcover_, Jan 28 2015
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