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A182304 G.f. satisfies: A(x) = x + d/dx A(x)^3/3.

%I A182304 #9 Jul 27 2014 16:30:11
%S A182304 1,1,4,25,200,1890,20248,240069,3102000,43226590,644778520,
%T A182304 10237504810,172274940656,3061868388100,57310884482160,
%U A182304 1126953452418765,23230411494361920,501015558669312390,11285603304699928600,265077311648541268030,6482435029801210815280
%N A182304 G.f. satisfies: A(x) = x + d/dx A(x)^3/3.
%H A182304 Vaclav Kotesovec, <a href="/A182304/b182304.txt">Table of n, a(n) for n = 1..440</a>
%F A182304 G.f. satisfies: A(x) = x + A'(x)*A(x)^2.
%F A182304 a(n) ~ c * n! * n^3, where c = 0.0147556981601927885048672335828437... - _Vaclav Kotesovec_, Feb 22 2014
%F A182304 G.f.: A(x) = Series_Reversion(x/F(x)) where F(x) = 1 + x*F(x)^3/(F(x) - x*F'(x)) is the g.f. of A245118. - _Paul D. Hanna_, Jul 27 2014
%e A182304 G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 200*x^5 + 1890*x^6 + 20248*x^7 +...
%e A182304 Related expansions.
%e A182304 A'(x) = 1 + 2*x + 12*x^2 + 100*x^3 + 1000*x^4 + 11340*x^5 + 141736*x^6 +...
%e A182304 A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 58*x^5 + 466*x^6 + 4380*x^7 + 46501*x^8 +...
%e A182304 A(x)^3 = x^3 + 3*x^4 + 15*x^5 + 100*x^6 + 810*x^7 + 7593*x^8 + 80023*x^9 +...
%e A182304 d/dx A(x)^3 = 3*x^2 + 12*x^3 + 75*x^4 + 600*x^5 + 5670*x^6 + 60744*x^7 +...
%o A182304 (PARI) {a(n)=local(A=x+x^2+x*O(x^n));for(i=1,n,A=x+deriv(A^3/3+O(x^31)));polcoeff(A,n)}
%o A182304 for(n=1,25,print1(a(n),", "))
%o A182304 (PARI) /* From g.f. F(x) of A245118 (_Paul D. Hanna_, Jul 27 2014): */
%o A182304 {a(n)=local(F=1+x); for(i=1, n, F = 1 + x*F^3/(F - x*F' +x*O(x^n))); polcoeff(serreverse(x/F), n)}
%o A182304 for(n=1, 25, print1(a(n), ", "))
%Y A182304 Cf. A245118, A088716.
%K A182304 nonn
%O A182304 1,3
%A A182304 _Paul D. Hanna_, Apr 23 2012