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%I #26 Nov 30 2020 08:42:23
%S 1,0,7,4,2,5,8,2,5,2,9,5,4,7,8,9,2,2,5,8,9,4,1,1,9,6,7,7,6,2,4,3,6,6,
%T 8,3,0,1,6,3,0,4,2,6,1,6,3,6,0,6,7,5,3,7,9,5,1,6,4,5,8,4,3,9,6,8,7,3,
%U 7,2,8,3,6,6,9,6,1,0,0,9,2,3,3,8,9,8,9,6,9,5,6,3,1,9,9,6,7,3,8,6,9,6
%N Decimal expansion of gamma_1(1/6), the first generalized Stieltjes constant at 1/6 (negated).
%H G. C. Greubel, <a href="/A254349/b254349.txt">Table of n, a(n) for n = 2..5002</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://dx.doi.org/10.1007/s11139-015-9763-z">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] (6*(-2*arctan(6*x) + 6*x*log(1/36 + x^2)))/((-1 + exp(2*Pi*x))*(1 + 36*x^2)) dx - (3 + log(6)/2)*log(6).
%e -10.742582529547892258941196776243668301630426163606753795...
%p evalf(int((6*(-2*arctan(6*x) + 6*x*log(1/36 + x^2)))/((-1 + exp(2*Pi*x))*(1 + 36*x^2)), x = 0..infinity) - (3 + log(6)/2)*log(6), 120); # _Vaclav Kotesovec_, Jan 29 2015
%t gamma1[1/6] = (1/2)*((-Log[6])*Log[24] - EulerGamma*Log[432] - 2*Log[2]*Log[2*Pi^2] + Log[(2*Pi)/Sqrt[3]]*Log[144*Pi^2] + Log[Pi]*Log[4/Gamma[1/6]^2] - 2*Log[12]*Log[Gamma[1/6]] - 2*Log[12*Pi]*Log[Gamma[5/6]] - Sqrt[3]*Pi*(EulerGamma + Log[(12*2^(2/3)*Pi^(3/2)*Gamma[5/6])/Gamma[1/6]^2]) + 2*StieltjesGamma[1] + Derivative[2, 0][Zeta][0, 1/6] - Derivative[2, 0][Zeta][0, 1/3] - 2*Derivative[2, 0][Zeta][0, 1/2] - Derivative[2, 0][Zeta][0, 2/3] + Derivative[2, 0][Zeta][0, 5/6]) // Re; RealDigits[gamma1[1/6], 10, 102] // First
%t (* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 1/6], 10, 102] // First
%Y Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254331 (gamma_1(1/3)), A254345 (gamma_1(2/3)), A254347 (gamma_1(1/4)), A254348 (gamma_1(3/4)), 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 2,3
%A _Jean-François Alcover_, Jan 29 2015
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