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%I #29 Nov 30 2020 08:42:08
%S 8,0,3,0,2,0,5,5,1,1,0,3,5,9,7,6,8,8,7,6,2,7,8,9,1,3,4,6,6,5,1,0,3,4,
%T 8,5,3,9,9,8,6,3,8,6,9,5,2,7,4,3,7,6,8,1,0,5,4,5,3,1,6,6,6,6,1,7,7,5,
%U 3,8,1,6,4,0,6,8,9,8,5,6,2,5,1,7,7,5,0,8,0,6,2,5,4,9,9,4,3,0,8,4,4,1,5,4,6
%N Decimal expansion of gamma_1(1/5), the first generalized Stieltjes constant at 1/5 (negated).
%H G. C. Greubel, <a href="/A251866/b251866.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://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>
%e -8.03020551103597688762789134665103485399863869527437681...
%t gamma1[1/5] = StieltjesGamma[1] + (1/2)*Sqrt[5]*(Derivative[2, 0][Zeta][0, 1/5] + Derivative[2, 0][Zeta][0, 4/5]) + (1/2)*(Pi*Sqrt[10 + 2*Sqrt[5]])*LogGamma[1/5] + (1/2)*(Pi*Sqrt[10 - 2*Sqrt[5]])*LogGamma[2/5] + ((1/2)*Sqrt[5]*Log[2] - 1/2)* Sqrt[5] *Log[1 + Sqrt[5]] - (1/10)*Pi*Sqrt[25 + 10*Sqrt[5]] -(5*Log[5])/4) *EulerGamma - (1/2)*Sqrt[5]*(Log[2] + Log[5] + Log[Pi] + (1/10)*Sqrt[25 - 10*Sqrt[5]] *Pi)*Log[1 + Sqrt[5]] + (1/2)*Sqrt[5]*Log[2]^2 + (1/8)*(Sqrt[5]*(1 - Sqrt[5]))*Log[5]^2 + ((3*Sqrt[5]) /4) *Log[2]*Log[5] + (Sqrt[5]/2)*Log[2]*Log[Pi] + (Sqrt[5]/4) *Log[5] *Log[Pi] - ((Pi*(2*Sqrt[25 + 10*Sqrt[5]] + 5*Sqrt[25 + 2*Sqrt[5]]))/20)*Log[2] -((Pi*(4*Sqrt[25 + 10*Sqrt[5]] - 5*Sqrt[5 + 2*Sqrt[5]]))/40)*Log[5] - ((Pi*(5*Sqrt[5 + 2*Sqrt[5]] + Sqrt[25 + 10*Sqrt[5]]))/10)*Log[Pi] // Re; RealDigits[gamma1[1/5], 10, 105] // First
%t (* or, from version 7 up: *) RealDigits[StieltjesGamma[1, 1/5], 10, 105] // 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)), A254349 (gamma_1(1/6)), A254350 (gamma_1(5/6)).
%K nonn,cons,easy
%O 1,1
%A _Jean-François Alcover_, Feb 16 2015
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