A242613 Decimal expansion of the sum of the alternating series tau(5), with tau(n) = Sum_{k>0} (-1)^k*log(k)^n/k.

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, 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, 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

Offset: 0

Author

Jean-François Alcover, May 19 2014

Keywords

• nonn
• cons

Examples

			-0.02451490765640978290742280068613711...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, chapter 2.21, p. 168.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000

Crossrefs

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

Programs

Mathematica

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]

Formula

tau(n) = -log(2)^(n+1)/(n+1) + Sum_(k=0..n-1) (binomial(n, k)*log(2)^(n-k)*gamma(k)).

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).