%I #24 Sep 03 2024 15:03:01

%S 1,-2,-2,22,94,-890,-9170,67030,1495870,-6581210,-362016050,194447350,

%T 120002960350,554823694150,-51277487618450,-601106981110250,

%U 26775789844186750,591304973974171750,-16113120605399179250

%N Polylogarithm li(-n,-2/3) multiplied by (5^(n+1))/3.

%C See the sequence A212846 which describes the general case of li(-n,-p/q). This sequence is obtained for p=2,q=3.

%H Stanislav Sykora, <a href="/A212847/b212847.txt">Table of n, a(n) for n = 0..100</a>

%F See formula in A212846, setting p=2,q=3.

%F From _Vaclav Kotesovec_, May 17 2022: (Start)

%F E.g.f.: 5/(3 + 2*exp(5*x)).

%F a(n) ~ n! * 2*(log(3/2) * cos(n*arctan(Pi/log(3/2))) - Pi * sin(n*arctan(Pi/log(3/2)))) * 5^(n+1) / (3 * (Pi^2 + log(3/2)^2)^(1 + n/2)). (End)

%e polylog(-5,-2/3)*5^6/3 = -890.

%t f[n_] := PolyLog[-n, -2/3] 5^(n + 1)/3; f[0] = 1; Array[f, 20, 0] (* _Robert G. Wilson v_, Dec 25 2015 *)

%o (PARI) \\ See A212846; run limnpq(nmax,2,3)

%Y Cf. A212846 (li(-n,-1/2)), A210246 (li(-n,-1/3)).

%Y Cf. A213127 through A213157.

%K sign

%O 0,2

%A _Stanislav Sykora_, May 28 2012