A242613 - OEIS

%I #19 Aug 30 2018 22:13:46

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

%T 1,0,2,8,7,5,7,0,7,0,9,2,3,7,9,1,5,0,3,7,4,2,9,0,5,1,1,2,7,2,9,8,3,7,

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

%N Decimal expansion of the sum of the alternating series tau(5), 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="/A242613/b242613.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(5) = gamma*log(2)^5 - (1/6)*log(2)^6 + 5*log(2)^4*gamma(1) + 10*log(2)^3*gamma(2) + 10*log(2)^2*gamma(3) + 5*log(2)*gamma(4).

%e -0.02451490765640978290742280068613711...

%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[5], 10, 100] // First]

%Y Cf. A001620, A082633, A086279, A086280, A086281, A242494, A242611, A242612.

%K nonn,cons

%O 0,2

%A _Jean-François Alcover_, May 19 2014