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A200319 E.g.f. satisfies: A(x) = x-1 + exp(A(x)^2).

%I A200319 #14 Jan 10 2014 17:04:32
%S A200319 1,2,12,132,2040,40440,979440,28034160,925858080,34654465440,
%T A200319 1449705660480,67029745527360,3394417068282240,186842736763562880,
%U A200319 11107390768144070400,709223357051739129600,48408150749346010022400,3517279496138031162739200,271050342684747077612160000
%N A200319 E.g.f. satisfies: A(x) = x-1 + exp(A(x)^2).
%F A200319 E.g.f.: Series_Reversion(1+x - exp(x^2)).
%F A200319 E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x^2)-1)^n / n!.
%F A200319 E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x^2)-1)^n/x / n! ).
%F A200319 a(n) ~ (c/2)^(1/4) * n^(n-1) / (sqrt(1+c) * exp(n) * (1+sqrt(c/2)-1/sqrt(2*c))^(n-1/2)), where c = LambertW(1/2) = 0.351733711249195826... (see A202356). - _Vaclav Kotesovec_, Jan 10 2014
%e A200319 E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2040*x^5/5! +...
%e A200319 where A(1+x - exp(x^2)) = x and A(x) = x-1 + exp(A(x)^2).
%t A200319 Rest[CoefficientList[InverseSeries[Series[1 - E^x^2 + x,{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 10 2014 *)
%o A200319 (PARI) {a(n)=n!*polcoeff(serreverse(1+x-exp(x^2+x^2*O(x^n))),n)}
%o A200319 for(n=1,25,print1(a(n),", "))
%o A200319 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A200319 {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, (exp(x^2+x*O(x^n))-1)^m)/m!); n!*polcoeff(A, n)}
%o A200319 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A200319 {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, (exp(x^2+x*O(x^n))-1)^m/x)/m!)+x*O(x^n)); n!*polcoeff(A, n)}
%Y A200319 Cf. A200320, A213640, A202356.
%K A200319 nonn
%O A200319 1,2
%A A200319 _Paul D. Hanna_, Nov 15 2011