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A242612 Decimal expansion of the sum of the alternating series tau(4), with tau(n) = Sum_{k>0} (-1)^k*log(k)^n/k. 3

%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

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)