A254349 Decimal expansion of gamma_1(1/6), the first generalized Stieltjes constant at 1/6 (negated).

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, 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, 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

Offset

2,3

Links

G. C. Greubel, Table of n, a(n) for n = 2..5002

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

Formula

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).

Example

-10.742582529547892258941196776243668301630426163606753795...

Maple

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

Mathematica

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
(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 1/6], 10, 102] // First

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)), A254350 (gamma_1(5/6)), A251866 (gamma_1(1/5)), A255188 (gamma_1(1/8)), A255189 (gamma_1(1/12)).

Sequence in context: A373018 A165244 A198356 * A019608 A245055 A335020

Adjacent sequences: A254346 A254347 A254348 * A254350 A254351 A254352

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jan 29 2015

Status

approved