A254350 Decimal expansion of gamma_1(5/6), the first generalized Stieltjes constant at 5/6 (negated).

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

Offset: 0

Author

Jean-François Alcover, Jan 29 2015

Keywords

• nonn
• cons
• easy

Examples

			-0.24616900381139073314849171532749069577086909012844...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments, arXiv:1401.3724 [math.NT], 2015.
• Iaroslav V. Blagouchine, A theorem ... (same title), Journal of Number Theory Volume 148, March 2015, Pages 537-592.
• Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal October 2014, Volume 35, Issue 1, pp 21-110.
• Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals: Full PDF text.
• Eric Weisstein's World of Mathematics, Hurwitz Zeta Function.
• Eric Weisstein's World of Mathematics, Stieltjes Constants.
• Wikipedia, Stieltjes constants

Crossrefs

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)), A251866 (gamma_1(1/5)), A255188 (gamma_1(1/8)), A255189 (gamma_1(1/12)).

Programs

Mathematica

gamma1[5/6] = (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] - *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[5/6], 10, 104] // First
(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 5/6], 10, 104] // First

Formula

Equals integral_[0..infinity] (6*(-10*arctan((6*x)/5) + 6*x*log(25/36 + x^2)))/((-1 + e^(2*Pi*x))*(25 + 36*x^2)) dx -(3/5 + (1/2)*log(6/5))*log(6/5).