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%I #22 Dec 14 2022 08:58:37
%S 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,
%T 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,
%U 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
%N Decimal expansion of 4th Stieltjes constant gamma_4.
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 166.
%H G. C. Greubel, <a href="/A086281/b086281.txt">Table of n, a(n) for n = 0..10000</a>
%H Krzysztof Maślanka and Andrzej Koleżyński, <a href="https://arxiv.org/abs/2210.04609">The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm</a>, arXiv preprint (2022). arXiv:2210.04609 [math.NT]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StieltjesConstants.html">Stieltjes Constants</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Stieltjes_constants">Stieltjes constants</a>
%F 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
%e 0.0023253...
%t Join[{0, 0}, RealDigits[ N[ -StieltjesGamma[4], 103]][[1]]] (* _Jean-François Alcover_, Nov 07 2012 *)
%Y Cf. A001620, A082633, A086279, A086280, A086282, A183141, A183167, A183206, A184853, A184854.
%K nonn,cons
%O 0,3
%A _Eric W. Weisstein_, Jul 14 2003
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