A086279 Decimal expansion of 2nd Stieltjes constant gamma_2 (negated).

0, 0, 9, 6, 9, 0, 3, 6, 3, 1, 9, 2, 8, 7, 2, 3, 1, 8, 4, 8, 4, 5, 3, 0, 3, 8, 6, 0, 3, 5, 2, 1, 2, 5, 2, 9, 3, 5, 9, 0, 6, 5, 8, 0, 6, 1, 0, 1, 3, 4, 0, 7, 4, 9, 8, 8, 0, 7, 0, 1, 3, 6, 5, 4, 5, 1, 8, 5, 0, 7, 5, 5, 3, 8, 2, 2, 8, 0, 4, 1, 4, 1, 7, 1, 9, 7, 8, 1, 9, 7, 3, 8, 1, 3, 7, 4, 5, 3, 7, 3, 1, 9

Offset: 0

Author

Eric W. Weisstein, Jul 14 2003

Keywords

• nonn
• cons

Examples

			-0.0096903...
		

References

• Steven 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, arXiv:2210.04609 [math.NT], 2022.
• Eric Weisstein's World of Mathematics, Stieltjes Constants
• Wikipedia, Stieltjes constants

Crossrefs

Cf. A001620, A082633, A086280, A086281, A086282, A183141, A183167, A183206, A184853, A184854.

Programs

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 *)

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