%I #17 Aug 30 2018 22:45:03
%S 0,1,7,9,9,6,9,3,8,1,0,6,8,9,1,4,0,7,7,9,5,3,6,7,8,2,1,4,3,6,1,5,2,6,
%T 2,3,8,9,8,1,1,2,3,4,5,1,3,9,0,2,3,3,4,9,2,9,4,5,0,2,4,7,9,9,9,1,3,2,
%U 2,5,6,2,4,6,3,8,0,8,5,8,4,3,0,9,4,2,9,7,0,5,9,1,9,5,1,4,2,4,2,9,9
%N Decimal expansion of the sum of the alternating series tau(4), with tau(n) = Sum_{k>0} (-1)^k*log(k)^n/k.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, chapter 2.21, p. 168.
%H G. C. Greubel, <a href="/A242612/b242612.txt">Table of n, a(n) for n = 0..10000</a>
%F tau(n) = -log(2)^(n+1)/(n+1) + Sum_(k=0..n-1) (binomial(n, k)*log(2)^(n-k)*gamma(k)).
%F tau(4) = gamma*log(2)^4 - (1/5)*log(2)^5 + 4*log(2)^3*gamma(1) + 6*log(2)^2*gamma(2) + 4*log(2)*gamma(3).
%e -0.017996938106891407795367821436152623898...
%t tau[n_] := -Log[2]^(n+1)/(n+1) + Sum[Binomial[n, k]*Log[2]^(n-k)*StieltjesGamma[k], {k, 0, n-1}]; Join[{0}, RealDigits[tau[4], 10, 100] // First]
%o (PARI) sumalt(k=1,(-1)^k*log(k)^4/k) \\ _Charles R Greathouse IV_, Mar 10 2016
%Y Cf. A001620, A082633, A086279, A086280, A242494, A242611, A242613.
%K nonn,cons
%O 0,3
%A _Jean-François Alcover_, May 19 2014