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%I #28 Dec 01 2020 03:29:14
%S 5,5,1,8,0,7,6,3,5,0,1,9,9,4,0,3,7,5,2,6,9,4,0,1,1,0,4,4,7,7,6,6,5,5,
%T 4,0,7,1,0,7,9,4,4,6,0,3,1,8,5,7,4,3,4,6,3,6,1,4,2,9,4,5,2,4,8,6,0,2,
%U 1,9,3,0,7,7,8,5,0,7,0,3,8,7,0,6,9,7,0,8,4,1,9,4,9,9,0,3,7,4,8,0,1,5,5
%N Decimal expansion of gamma_1(1/4), the first generalized Stieltjes constant at 1/4 (negated).
%H G. C. Greubel, <a href="/A254347/b254347.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="https://doi.org/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] (4*(-2*arctan(4*x) + 4*x*log(1/16 + x^2)))/((-1 + exp(2*Pi*x))*(1 + 16*x^2)) dx - (2 + log(4)/2)*log(4).
%e -5.5180763501994037526940110447766554071079446031857434636...
%p evalf(int((4*(-2*arctan(4*x)+4*x*log(1/16+x^2)))/((-1+exp(2*Pi*x))*(16*x^2+1)), x = 0..infinity) - (2+(1/2)*log(4))*log(4), 120); # _Vaclav Kotesovec_, Jan 29 2015
%t gamma1[1/4] = -1/2*Log[4]^2 - 1/2*EulerGamma*(Pi + Log[64]) - Log[4]*Log[2*Pi] - Log[2*Pi]^2 + Log[Pi]*Log[8*Pi] - 1/2*Pi*Log[8*Pi*Gamma[3/4]^2/Gamma[1/4]^2] + StieltjesGamma[1] - Derivative[2, 0][Zeta][0, 1/2] // Re; RealDigits[gamma1[1/4], 10, 103] // First
%t (* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 1/4], 10, 103] // First
%Y Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254331 (gamma_1(1/3)), A254345 (gamma_1(2/3)), 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 29 2015
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