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%I #28 May 01 2023 10:01:33
%S 1,2,7,37,273,2661,32773,491555,8715409,178438681,4142334501,
%T 107483043735,3081956918857,96759352320437,3300826000845493,
%U 121569984050610331,4807542796319581089,203167758634027130289
%N Expansion of e.g.f. LambertW(-x)/(x*(x-1)).
%H Vincenzo Librandi, <a href="/A102743/b102743.txt">Table of n, a(n) for n = 0..100</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F a(n) = n!*Sum_{k=1..n+1} k^(k-1)/k!. - _Vladeta Jovovic_, Oct 17 2007
%F a(n) ~ exp(2)/(exp(1)-1) * n^(n-1). - _Vaclav Kotesovec_, Nov 27 2012
%F E.g.f.: W(0)/(2-2*x) , where W(k) = 1 + 1/( 1 - x*(k+2)^k/( x*(k+2)^k + (k+1)^k/W(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 19 2013
%F From _Seiichi Manyama_, May 01 2023: (Start)
%F E.g.f.: exp(-LambertW(-x))/(1-x).
%F a(0) = 1; a(n) = n*a(n-1) + (n+1)^(n-1). (End)
%t CoefficientList[Series[LambertW[-x]/(x*(x-1)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Nov 27 2012 *)
%o (PARI) my(x='x+O('x^50)); Vec(serlaplace(lambertw(-x)/(x*(x-1)))) \\ _G. C. Greubel_, Nov 08 2017
%Y Cf. A277506.
%K easy,nonn
%O 0,2
%A _Vladeta Jovovic_, Feb 08 2005
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