A086281 Decimal expansion of 4th Stieltjes constant gamma_4.

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

Offset: 0

Author

Eric W. Weisstein, Jul 14 2003

Keywords

• nonn
• cons

Examples

			0.0023253...
		

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, A086279, A086280, A086282, A183141, A183167, A183206, A184853, A184854.

Programs

Mathematica

Join[{0, 0}, RealDigits[ N[ -StieltjesGamma[4], 103]][[1]]] (* Jean-François Alcover, Nov 07 2012 *)

Formula

Using the abbreviations a = log(z^2 + 1/4)/2, b = arctan(2*z) and c = cosh(Pi*z) then gamma_4 = -(Pi/5)*Integral_{0..infinity} (a^5-10*a^3*b^2+5*a*b^4)/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