A251866 Decimal expansion of gamma_1(1/5), the first generalized Stieltjes constant at 1/5 (negated).

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, 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, 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

Offset

1,1

Links

G. C. Greubel, Table of n, a(n) for n = 1..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

Example

-8.03020551103597688762789134665103485399863869527437681...

Mathematica

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
(* or, from version 7 up: *) RealDigits[StieltjesGamma[1, 1/5], 10, 105] // 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)), A254349 (gamma_1(1/6)), A254350 (gamma_1(5/6)).

Sequence in context: A200093 A179068 A144455 * A300713 A373513 A020837

Adjacent sequences: A251863 A251864 A251865 * A251867 A251868 A251869

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Feb 16 2015

Status

approved