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%I A138013 #14 Dec 23 2014 15:31:15
%S A138013 1,1,3,17,146,1694,24834,440586,9180800,219829536,5948287560,
%T A138013 179508872520,5978006444112,217772950035120,8614798644364080,
%U A138013 367768502385434640,16852524904388586240,825075552824125305600,42981992589364756939008,2373967488394457834095872
%N A138013 E.g.f. satisfies: A(x) = 1 - log(1 - x*A(x)).
%C A138013 a(n) = A038037(n+1)/(n+1) for n>=0 where A038037(n) is the number of labeled rooted compound windmills (mobiles) with n nodes.
%F A138013 E.g.f.: A(x) = (1/x)*Series_Reversion[ x/(1 - log(1-x)) ].
%F A138013 E.g.f.: A(x) = 1 + Series_Reversion( (1-exp(-x))/(1+x) ).
%F A138013 E.g.f. A(x) satisfies: exp(1 - A(x)) = 1 - x*A(x).
%F A138013 a(n) ~ sqrt(-1-LambertW(-1,-exp(-2))) * (-LambertW(-1,-exp(-2)))^n * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Dec 27 2013
%F A138013 a(n) = sum(n!/(n+1-k)! * |stirling1(n,k)|, k=0..n). - _Michael D. Weiner_, Dec 23 2014
%e A138013 E.g.f.: A(x) = 1 + x + 3x^2/2! + 17x^3/3! + 146x^4/4! + 1694x^5/5! + ...
%e A138013 where A(x) = 1 - log(1 - x*A(x)):
%e A138013 A(x) = 1 + x*A(x) + x^2*A(x)^2/2 + x^3*A(x)^3/3 +...+ x^n*A(x)^n/n +...
%t A138013 CoefficientList[1 + InverseSeries[Series[(1-E^(-x))/(1+x), {x, 0, 20}], x],x] * Range[0, 20]! (* _Vaclav Kotesovec_, Dec 27 2013 *)
%o A138013 (PARI) {a(n)=n!*polcoeff(1/x*serreverse(x/(1-log(1-x + x*O(x^n) ))),n+1)}
%o A138013 (PARI) {a(n)=n!*polcoeff(1 + serreverse((1-exp(-x+x^2*O(x^n)))/(1+x +x*O(x^n))),n)}
%Y A138013 Cf. A038037.
%K A138013 nonn
%O A138013 0,3
%A A138013 _Paul D. Hanna_, Feb 27 2008

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