%I #12 Jun 19 2015 05:03:43

%S 1,2,14,182,3526,91422,2978910,117081974,5393393078,285072735950,

%T 17009730803086,1131081110962662,82949497319012070,

%U 6651426091458349502,578967663130916841662,54369741954121640179286,5479228772620109128533526,589841997033535953174559662

%N E.g.f.: (1/x) * Series_Reversion( -x + 2*x*exp(-x) ).

%F E.g.f.: 1 + (1/x)*Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (1 - exp(-x))^n * x^n / n!.

%F E.g.f.: exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (1 - exp(-x))^n * x^(n-1) / n! ).

%F a(n) = A259062(n+1) / (n+1). - _Vaclav Kotesovec_, Jun 19 2015

%F a(n) ~ (1-c) * n^(n-1) / (sqrt(1+c) * (c + 1/c - 2)^(n+1) * exp(n)), where c = LambertW(exp(1)/2) = 0.685076942154593946... . - _Vaclav Kotesovec_, Jun 19 2015

%e E.g.f.: A(x) = 1 + 2*x + 14*x^2/2! + 182*x^3/3! + 3526*x^4/4! + 91422*x^5/5! +...

%t CoefficientList[1/x*InverseSeries[Series[-x + 2*x*E^(-x), {x, 0, 21}], x],x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jun 19 2015 *)

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

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n)=local(A=x); A = 1 + (1/x)*sum(m=1, n, Dx(m-1, 2^m*(1-exp(-x+x^2*O(x^n)))^m*x^m/m!)); n!*polcoeff(A, n)}

%o for(n=0, 25, print1(a(n), ", "))

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n)=local(A=x+x^2+x*O(x^n)); A = exp(sum(m=1, n+1, 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 for(n=0, 25, print1(a(n), ", "))

%Y Cf. A259062.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 18 2015