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A254347 Decimal expansion of gamma_1(1/4), the first generalized Stieltjes constant at 1/4 (negated). 10
5, 5, 1, 8, 0, 7, 6, 3, 5, 0, 1, 9, 9, 4, 0, 3, 7, 5, 2, 6, 9, 4, 0, 1, 1, 0, 4, 4, 7, 7, 6, 6, 5, 5, 4, 0, 7, 1, 0, 7, 9, 4, 4, 6, 0, 3, 1, 8, 5, 7, 4, 3, 4, 6, 3, 6, 1, 4, 2, 9, 4, 5, 2, 4, 8, 6, 0, 2, 1, 9, 3, 0, 7, 7, 8, 5, 0, 7, 0, 3, 8, 7, 0, 6, 9, 7, 0, 8, 4, 1, 9, 4, 9, 9, 0, 3, 7, 4, 8, 0, 1, 5, 5 (list; constant; graph; refs; listen; history; text; internal format)
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;

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] (4*(-2*arctan(4*x) + 4*x*log(1/16 + x^2)))/((-1 + exp(2*Pi*x))*(1 + 16*x^2)) dx - (2 + log(4)/2)*log(4).

EXAMPLE

-5.5180763501994037526940110447766554071079446031857434636...

MAPLE

evalf(int((4*(-2*arctan(4*x)+4*x*log(1/16+x^2)))/((-1+exp(2*Pi*x))*(16*x^2+1)), x = 0..infinity) - (2+(1/2)*log(4))*log(4), 120); # Vaclav Kotesovec, Jan 29 2015

MATHEMATICA

gamma1[1/4] = -1/2*Log[4]^2 - 1/2*EulerGamma*(Pi + Log[64]) - Log[4]*Log[2*Pi] - Log[2*Pi]^2 + Log[Pi]*Log[8*Pi] - 1/2*Pi*Log[8*Pi*Gamma[3/4]^2/Gamma[1/4]^2] + StieltjesGamma[1] - Derivative[2, 0][Zeta][0, 1/2] // Re; RealDigits[gamma1[1/4], 10, 103] // First

(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 1/4], 10, 103] // First

CROSSREFS

Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254331 (gamma_1(1/3)), A254345 (gamma_1(2/3)), 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: A014287 A154945 A300710 * A011094 A204005 A075298

Adjacent sequences:  A254344 A254345 A254346 * A254348 A254349 A254350

KEYWORD

nonn,cons,easy

AUTHOR

Jean-François Alcover, Jan 29 2015

STATUS

approved

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Last modified August 21 10:21 EDT 2018. Contains 313937 sequences. (Running on oeis4.)