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%I #13 Jan 12 2014 11:15:32
%S 1,2,11,126,2049,42012,1047507,30867540,1049597685,40441973328,
%T 1741357779039,82865509846776,4318613855629605,244629863660429712,
%U 14965278826983897303,983295107764013223504,69061868853286423944249,5163430410995824208371968
%N E.g.f. A(x) satisfies: exp(A(x)) = x + exp(3*A(x)^2/2), with A(0) = 0.
%F E.g.f.: A(x) = Series_Reversion( exp(x) - exp(3*x^2/2) ).
%F a(n) ~ sqrt((3*s - 1)/(3 - 3*s + 9*s^2)) * n^(n-1) / (exp(s+1)-exp(3*s^2/2+1))^n, where s = 0.39177456704014800117... is the root of the equation 3*s*exp(3*s^2/2 - s) = 1. - _Vaclav Kotesovec_, Jan 12 2014
%e E.g.f.: A(x) = x + 2*x^2/2! + 11*x^3/3! + 126*x^4/4! + 2049*x^5/5! +...
%e where A( exp(x) - exp(3*x^2/2) ) = x.
%e Related expansions:
%e exp(A(x)) = 1 + x + 3*x^2/2! + 18*x^3/3! + 195*x^4/4! + 3090*x^5/5! +...
%e exp(3*A(x)^2/2) = 1 + 3*x^2/2! + 18*x^3/3! + 195*x^4/4! + 3090*x^5/5! +...
%t Rest[CoefficientList[InverseSeries[Series[E^x - E^(3*x^2/2),{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 12 2014 *)
%o (PARI) {a(n)=local(X=x+x*O(x^n));if(n<1, 0, n!*polcoeff(serreverse(exp(X)-exp(3*X^2/2)), n))}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A138014, A206402, A206403, A206404, A206405.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Feb 07 2012