A242611 Decimal expansion of the sum of the alternating series tau(3), with tau(n) = Sum_{k>0} (-1)^k*log(k)^n/k.

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

Offset

0,3

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

Formula

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

tau(3) = gamma*log(2)^3 - (1/4)*log(2)^4 + 3*log(2)^2*gamma(1) + 3*log(2)*gamma(2).

Example

0.009413950232493089735171955362333...

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, 0}, RealDigits[tau[3], 10, 100] // First]

Prog

(PARI) -suminf(k=1, (-1)^k*log(k)^n/k) \\ Charles R Greathouse IV, Mar 10 2016

Crossrefs

Cf. A001620, A082633, A086279, A242494, A242612, A242613.

Sequence in context: A010158 A336274 A286229 * A098454 A298531 A154206

Adjacent sequences: A242608 A242609 A242610 * A242612 A242613 A242614

Keyword

nonn,cons

Author

Jean-François Alcover, May 19 2014

Status

approved