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%I A259062 #13 Jun 19 2015 04:54:25
%S A259062 1,4,42,728,17630,548532,20852370,936655792,48540537702,2850727359500,
%T A259062 187107038833946,13572973331551944,1078343465147156910,
%U A259062 93119965280416893028,8684514946963752624930,869915871265946242868576,93146889134541855185069942,10617155946603647157142073916
%N A259062 E.g.f.: Series_Reversion( -x + 2*x*exp(-x) ).
%F A259062 E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (1 - exp(-x))^n * x^n / n!.
%F A259062 E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (1 - exp(-x))^n * x^(n-1) / n! ).
%F A259062 a(n) ~ (1-c) * n^(n-1) / (sqrt(1+c) * (c + 1/c - 2)^n * exp(n)), where c = LambertW(exp(1)/2) = 0.685076942154593946... . - _Vaclav Kotesovec_, Jun 19 2015
%e A259062 E.g.f.: A(x) = x + 4*x^2/2! + 42*x^3/3! + 728*x^4/4! + 17630*x^5/5! +...
%e A259062 where A(-x + 2*x*exp(-x)) = x.
%t A259062 Rest[CoefficientList[InverseSeries[Series[-x + 2*x*E^(-x), {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jun 19 2015 *)
%o A259062 (PARI) {a(n) = local(A=x); A = serreverse(-x + 2*x*exp(-x +x*O(x^n) )); n!*polcoeff(A, n)}
%o A259062 for(n=1, 20, print1(a(n), ", "))
%o A259062 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A259062 {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, 2^m*(1-exp(-x+x*O(x^n)))^m*x^m/m!)); n!*polcoeff(A, n)}
%o A259062 for(n=1, 25, print1(a(n), ", "))
%o A259062 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A259062 {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, 2^m*(1-exp(-x+x*O(x^n)))^m*x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
%o A259062 for(n=1, 25, print1(a(n), ", "))
%Y A259062 Cf. A258872, A053492, A259063, A259064, A259065, A259066.
%K A259062 nonn
%O A259062 1,2
%A A259062 _Paul D. Hanna_, Jun 18 2015

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