A302581 - OEIS

%I #9 Jun 09 2019 03:58:43

%S 0,1,-3,20,-186,2249,-33360,586172,-11901008,274098393,-7060189120,

%T 201092672604,-6275340884736,212915635727313,-7803567334571008,

%U 307245946117223700,-12933084380738398208,579587518114690731601,-27550568677612746940416,1384553892443352890245636

%N a(n) = n! * [x^n] -exp(-n*x)*log(1 - x).

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Lo#logarithmic">Index entries for sequences related to logarithmic numbers</a>

%F a(n) = Sum_{k=1..n} (-n)^(n-k)*(k-1)!*binomial(n,k).

%F E.g.f.: -log(1 - LambertW(x))/(1 + LambertW(x)). - _Vaclav Kotesovec_, Jun 09 2019

%F a(n) ~ -(-1)^n * log(2) * n^n. - _Vaclav Kotesovec_, Jun 09 2019

%t Table[n! SeriesCoefficient[-Exp[-n x] Log[1 - x], {x, 0, n}], {n, 0, 19}]

%t Table[Sum[(-n)^(n - k) (k - 1)! Binomial[n, k], {k, 1, n}], {n, 0, 19}]

%t nmax = 20; CoefficientList[Series[-Log[1 - LambertW[x]]/(1 + LambertW[x]), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jun 09 2019 *)

%Y Cf. A002104, A002741, A104150, A134095, A190314.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Apr 10 2018