A255189 - OEIS

%I #30 Jan 17 2017 02:36:42

%S 2,9,8,4,2,8,7,8,2,3,2,0,4,1,3,3,1,3,0,3,3,5,1,0,2,0,2,6,0,7,5,9,2,6,

%T 3,2,3,9,8,9,2,0,4,4,0,0,1,8,6,1,0,0,5,6,8,7,0,3,6,1,0,6,7,8,3,0,9,3,

%U 3,3,8,8,5,1,5,6,1,2,3,1,6,1,4,6,4,6,2,5,1,2,7,6,9,7,0,1,2,4,2,3,4,8,7,8

%N Decimal expansion of gamma_1(1/12), the first generalized Stieltjes constant at 1/12 (negated).

%H G. C. Greubel, <a href="/A255189/b255189.txt">Table of n, a(n) for n = 2..5002</a>

%H Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jnt.2014.08.009">A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations</a>, Journal of Number Theory (Elsevier), Volume 148, pages 537-592, March 2015 (<a href="http://arxiv.org/abs/1401.3724">arXiv preprint</a>).

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>.

%H Eric Weisstein's MathWorld, <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 -29.842878232041331303351020260759263239892044001861...

%t gamma1[1/12] = StieltjesGamma[1] + Sqrt[3]*(Derivative[2, 0][Zeta][0, 1/12] + Derivative[2, 0][Zeta][0, 11/12]) + 4*Pi*LogGamma[1/4] + 3*Pi*Sqrt[3]*LogGamma[1/3] - (((2 + Sqrt[3])/2)*Pi + (3/2)*Log[3] - Sqrt[3]*(1 - Sqrt[3])*Log[2] + 2*Sqrt[3]*Log[1 + Sqrt[3]])*EulerGamma - 2*Sqrt[3]*(3*Log[2] + Log[3] + Log[Pi])* Log[1 + Sqrt[3]] - ((7 - 6*Sqrt[3])/2)*Log[2]^2 - (3/4)*Log[3]^2 + 3*Sqrt[3]*((1 - Sqrt[3])/2)*Log[3]*Log[2] + Sqrt[3]*Log[2]*Log[Pi] - Pi*((17 + 8*Sqrt[3])/(2*Sqrt[3]))*Log[2] + ((Pi*(1 - Sqrt[3])*Sqrt[3])/4)*Log[3] - Pi*Sqrt[3]*(2 + Sqrt[3])*Log[Pi] // Re; RealDigits[gamma1[1/12], 10, 104] // First

%t (* or, from version 7 up: *) RealDigits[StieltjesGamma[1, 1/12], 10, 104] // 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)), A251866 (gamma_1(1/5)) A255188 (gamma_1(1/8)).

%K nonn,cons,easy

%O 2,1

%A _Jean-François Alcover_, Feb 16 2015