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A254345 Decimal expansion of gamma_1(2/3), the first generalized Stieltjes constant at 2/3 (negated). 10
5, 9, 8, 9, 0, 6, 2, 8, 4, 2, 8, 5, 9, 8, 9, 2, 9, 2, 5, 6, 7, 8, 7, 6, 0, 2, 1, 2, 6, 9, 2, 5, 0, 2, 5, 6, 6, 6, 3, 9, 1, 3, 4, 0, 7, 8, 1, 7, 5, 7, 1, 4, 9, 1, 5, 8, 6, 5, 0, 1, 5, 6, 9, 7, 1, 8, 7, 2, 0, 7, 6, 5, 0, 2, 5, 5, 0, 4, 7, 8, 6, 7, 7, 3, 3, 4, 2, 4, 7, 9, 8, 8, 1, 7, 2, 9, 0, 7, 1, 1, 1, 5, 2, 9, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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;

A theorem ... (same title), Journal of Number Theory Volume 148, March 2015, Pages 537-592;

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;

Rediscovery of Malmsten’s integrals: Full PDF text.

Eric Weisstein's MathWorld,

Hurwitz Zeta Function;

Stieltjes Constants.

Wikipedia, Stieltjes constants

FORMULA

Equals integral_[0..infinity] (-12*arctan(3*x/2) + 9*x*log(4/9 + x^2))/((-1 + exp(2*Pi*x))*(4 + 9*x^2)) dx - (3/4 + (1/2)*log(3/2))*log(3/2).

EXAMPLE

-0.5989062842859892925678760212692502566639134078175714915865...

MAPLE

evalf(int((-12*arctan(3*x*(1/2))+9*x*log(4/9+x^2))/((-1+exp(2*Pi*x))*(9*x^2+4)), x = 0..infinity) - (3/4+(1/2)*log(3/2))*log(3/2), 120); # Vaclav Kotesovec, Jan 29 2015

MATHEMATICA

gamma1[2/3] = (1/12)*(Sqrt[3]*Pi*(2*EulerGamma + Log[(576*Pi^5)/Gamma[1/6]^6]) - 6*(EulerGamma*Log[27] + Log[3]*Log[18*Pi] - 2*StieltjesGamma[1] + Derivative[2, 0][Zeta][0, 1/3] + Derivative[2, 0][Zeta][0, 2/3])) // Re; RealDigits[gamma1[2/3], 10, 105] // First

(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 2/3], 10, 105] // First

CROSSREFS

Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254331 (gamma_1(1/3)), 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)), A255189 (gamma_1(1/12)).

Sequence in context: A112678 A021171 A011493 * A077771 A019754 A244249

Adjacent sequences:  A254342 A254343 A254344 * A254346 A254347 A254348

KEYWORD

nonn,cons,easy

AUTHOR

Jean-François Alcover, Jan 29 2015

STATUS

approved

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Last modified February 21 22:56 EST 2018. Contains 299427 sequences. (Running on oeis4.)