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A215003 E.g.f. satisfies: A(x) = x + A(x)^2*exp(A(x))/2.

%I A215003 #6 Dec 28 2013 09:12:40
%S A215003 1,1,6,51,610,9360,175371,3881269,99083916,2866173795,92650139560,
%T A215003 3309877247886,129495296417565,5506601964331741,252882022593123210,
%U A215003 12473008516763238375,657621285983402967766,36908182475893682882532,2196924777991866630353571
%N A215003 E.g.f. satisfies: A(x) = x + A(x)^2*exp(A(x))/2.
%F A215003 E.g.f.: Series_Reversion(x - x^2*exp(x)/2).
%F A215003 E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) exp(n*x)/2^n*x^(2*n) / n!.
%F A215003 E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) exp(n*x)/2^n*x^(2*n-1) / n! ).
%F A215003 O.g.f.: Sum_{n>=0} (2*n)!/n!/2^n * x^(n+1) / (1 - n*x)^(2*n+1).
%F A215003 a(n) = Sum_{k=0..n-1} k^(n-k-1)/(n-k-1)! * (n+k-1)!/k!/2^k.
%F A215003 a(n) ~ r*sqrt((1+r)/(2+r*(4+r))) * n^(n-1) / (exp(n) * (r*(1+r)/(2+r))^n), where r = 0.4912251835444738797155054345... is the root of the equation r*(2+r)*exp(r) = 2. - _Vaclav Kotesovec_, Dec 28 2013
%e A215003 E.g.f.: A(x) = x + x^2/2! + 6*x^3/3! + 51*x^4/4! + 610*x^5/5! + 9360*x^6/6! +...
%e A215003 where A(x - x^2*exp(x)/2) = x and A(x) = x + A(x)^2*exp(A(x))/2.
%e A215003 Related expansions:
%e A215003 A(x)^2 = 2*x^2/2! + 6*x^3/3! + 54*x^4/4! + 630*x^5/5! + 9570*x^6/6! +...
%e A215003 exp(A(x)) = 1 + x + 2*x^2/2! + 10*x^3/3! + 85*x^4/4! + 1011*x^5/5! + 15466*x^6/6! + 289213*x^7/7! +...
%e A215003 A(x) = x + exp(x)*x^2/2 + d/dx exp(2*x)*x^4/2!/4 + d^2/dx^2 exp(3*x)*x^6/3!/8 + d^3/dx^3 exp(4*x)*x^8/4!/16 +...
%e A215003 log(A(x)/x) = exp(x)*x/2 + d/dx exp(2*x)*x^3/2!/4 + d^2/dx^2 exp(3*x)*x^5/3!/8 + d^3/dx^3 exp(4*x)*x^7/4!/16 +...
%e A215003 Ordinary Generating Function:
%e A215003 O.g.f.: x + x^2 + 6*x^3 + 51*x^4 + 610*x^5 + 9360*x^6 + 175371*x^7 +...
%e A215003 O.g.f.: x + 2*x^2/(1-x)^3/2 + 6*2!*x^3/(1-2*x)^5/4 + 20*3!*x^4/(1-3*x)^7/8 + 70*4!*x^5/(1-4*x)^9/16 + 252*5!*x^6/(1-5*x)^11/32 +...+ (2*n)!/n!*x^(n+1)/(1-n*x)^(2*n+1)/2^n +...
%t A215003 Rest[CoefficientList[InverseSeries[Series[x - x^2*E^x/2, {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Dec 28 2013 *)
%o A215003 (PARI) {a(n)=sum(k=0, n-1, k^(n-k-1)/(n-k-1)! * (n+k-1)!/k!/2^k )}
%o A215003 (PARI) {a(n)=n!*polcoeff(serreverse(x-x^2*exp(x+x*O(x^n))/2), n)}
%o A215003 for(n=1, 25, print1(a(n), ", "))
%o A215003 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A215003 {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, exp(m*x+x*O(x^n))/2^m*x^(2*m)/m!)); n!*polcoeff(A, n)}
%o A215003 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A215003 {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, exp(m*x+x*O(x^n))/2^m*x^(2*m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
%o A215003 (PARI) /* O.g.f.: */
%o A215003 {a(n)=polcoeff(sum(m=0, n, (2*m)!/m!/2^m*x^(m+1)/(1-m*x+x*O(x^n))^(2*m+1)), n)}
%Y A215003 Cf. A213643.
%K A215003 nonn
%O A215003 1,3
%A A215003 _Paul D. Hanna_, Jul 30 2012