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E.g.f. A(x) satisfies: exp(A(x)) = x + exp(2*A(x)^2), with A(0) = 0.
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%I #10 Jan 12 2014 11:15:36

%S 1,3,26,422,9684,284536,10205264,432507008,21149344320,1172055816864,

%T 72593488746624,4969455399927168,372585629959484928,

%U 30363657581135890176,2672420848359072517632,252632488649577779398656,25529176319428221234402816,2746226455049097879478060032

%N E.g.f. A(x) satisfies: exp(A(x)) = x + exp(2*A(x)^2), with A(0) = 0.

%F E.g.f.: A(x) = Series_Reversion( exp(x) - exp(2*x^2) ).

%F a(n) ~ 1/2 * sqrt((4*s - 1)/(1 - s + 4*s^2)) * n^(n-1) / (exp(s+1)-exp(2*s^2+1))^n, where s = 0.28268257266202691... is the root of the equation exp(s) = 4*exp(2*s^2)*s. - _Vaclav Kotesovec_, Jan 12 2014

%e E.g.f.: A(x) = x + 3*x^2/2! + 26*x^3/3! + 422*x^4/4! + 9684*x^5/5! +...

%e where A( exp(x) - exp(2*x^2) ) = x.

%e Related expansions:

%e exp(A(x)) = 1 + x + 4*x^2/2! + 36*x^3/3! + 572*x^4/4! + 13000*x^5/5! +...

%e exp(2*A(x)^2) = 1 + 4*x^2/2! + 36*x^3/3! + 572*x^4/4! + 13000*x^5/5! +...

%t Rest[CoefficientList[InverseSeries[Series[E^x - E^(2*x^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(2*X^2)), n))}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A138014, A206401, A206403, A206404, A206405.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Feb 07 2012