OFFSET
0,3
REFERENCES
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 166.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
Krzysztof Maślanka and Andrzej Koleżyński, The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm, arXiv preprint (2022). arXiv:2210.04609 [math.NT]
Eric Weisstein's World of Mathematics, Stieltjes Constants
Wikipedia, Stieltjes constants
FORMULA
Using the abbreviations a = log(z^2 + 1/4)/2, b = arctan(2*z) and c = cosh(Pi*z) then gamma_2 = -(Pi/3)*Integral_{0..infinity}(a^3-3*a*b^2)/c^2. The general case is for n >= 0 (which includes Euler's gamma as gamma_0) gamma_n = (-Pi/(n+1))* Integral_{0..infinity} sigma(n+1)/c^2, where sigma(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*b^(2*k)*a^(n-2*k). - Peter Luschny, Apr 19 2018
EXAMPLE
-0.0096903...
MAPLE
evalf(gamma(2)); # R. J. Mathar, Feb 02 2011
MATHEMATICA
Join[{0, 0}, RealDigits[N[-StieltjesGamma[2], 101]][[1]]] (* Jean-François Alcover, Oct 23 2012 *)
N[4*EulerGamma^3 + Residue[Zeta[s]^4 / 2 - 2*EulerGamma*Zeta[s]^3, {s, 1}], 100] (* Vaclav Kotesovec, Jan 07 2017 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jul 14 2003
STATUS
approved