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A138013 E.g.f. satisfies: A(x) = 1 - log(1 - x*A(x)). 11

%I #14 Dec 23 2014 15:31:15

%S 1,1,3,17,146,1694,24834,440586,9180800,219829536,5948287560,

%T 179508872520,5978006444112,217772950035120,8614798644364080,

%U 367768502385434640,16852524904388586240,825075552824125305600,42981992589364756939008,2373967488394457834095872

%N E.g.f. satisfies: A(x) = 1 - log(1 - x*A(x)).

%C 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 E.g.f.: A(x) = (1/x)*Series_Reversion[ x/(1 - log(1-x)) ].

%F E.g.f.: A(x) = 1 + Series_Reversion( (1-exp(-x))/(1+x) ).

%F E.g.f. A(x) satisfies: exp(1 - A(x)) = 1 - x*A(x).

%F a(n) ~ sqrt(-1-LambertW(-1,-exp(-2))) * (-LambertW(-1,-exp(-2)))^n * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Dec 27 2013

%F a(n) = sum(n!/(n+1-k)! * |stirling1(n,k)|, k=0..n). - _Michael D. Weiner_, Dec 23 2014

%e E.g.f.: A(x) = 1 + x + 3x^2/2! + 17x^3/3! + 146x^4/4! + 1694x^5/5! + ...

%e where A(x) = 1 - log(1 - x*A(x)):

%e 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 CoefficientList[1 + InverseSeries[Series[(1-E^(-x))/(1+x), {x, 0, 20}], x],x] * Range[0, 20]! (* _Vaclav Kotesovec_, Dec 27 2013 *)

%o (PARI) {a(n)=n!*polcoeff(1/x*serreverse(x/(1-log(1-x + x*O(x^n) ))),n+1)}

%o (PARI) {a(n)=n!*polcoeff(1 + serreverse((1-exp(-x+x^2*O(x^n)))/(1+x +x*O(x^n))),n)}

%Y Cf. A038037.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 27 2008

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