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

0,3

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

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

Sequence in context: A153071 A336085 A363539 * A155533 A083281 A019711

Adjacent sequences: A086276 A086277 A086278 * A086280 A086281 A086282

Keyword

nonn,cons

Author

Eric W. Weisstein, Jul 14 2003

Status

approved