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A234293 E.g.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x)^3 dx.

%I A234293 #15 Dec 26 2013 18:26:34
%S A234293 1,1,5,57,1053,27057,891765,35883369,1705399821,93486709089,
%T A234293 5806695485925,403035074738073,30915209058366717,2597006055407919633,
%U A234293 237114260894842246485,23379981536098259658441,2475981772053312305191149,280285314108547303508788161,33774891856931450901116286405
%N A234293 E.g.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x)^3 dx.
%F A234293 E.g.f.: 1 + Series_Reversion( 4*x/(1+x) - 3*log(1+x) ).
%F A234293 E.g.f.: 1 / ( d/dx Series_Reversion( Integral G(x)^3 dx ) )^(1/3), where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
%F A234293 a(n) ~ n! * (2/3)^(3/2) / (sqrt(Pi) * n^(3/2) * (1-6*log(2)+3*log(3))^(n-1/2)). - _Vaclav Kotesovec_, Dec 26 2013
%F A234293 E.g.f.: -4/(3*LambertW(-1,-4/3*exp((x-4)/3))). - _Vaclav Kotesovec_, Dec 26 2013
%e A234293 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 57*x^3/3! + 1053*x^4/4! + 27057*x^5/5! +...
%e A234293 where A(4*x/(1+x) - 3*log(1+x)) = 1+x.
%e A234293 Related series:
%e A234293 A(x)^3 = 1 + 3*x + 21*x^2/2! + 267*x^3/3! + 5157*x^4/4! + 135531*x^5/5! +...
%e A234293 A(x)^4 = 1 + 4*x + 32*x^2/2! + 432*x^3/3! + 8592*x^4/4! + 229488*x^5/5! +...
%e A234293 1/A(x)^3 = 1 - 3*x - 3*x^2/2! - 51*x^3/3! - 963*x^4/4! - 25011*x^5/5! +...
%e A234293 The series reversion of A(x)-1 begins:
%e A234293 4*x/(1+x) - 3*log(1+x) = x - 5*x^2/2 + 9*x^3/3 - 13*x^4/4 + 17*x^5/5 - 21*x^6/6 +...
%p A234293 seq(n! * coeff(series(-4/(3*LambertW(-1, -4/3*exp((x-4)/3))), x, n+1), x, n), n = 0..10) # _Vaclav Kotesovec_, Dec 26 2013
%t A234293 CoefficientList[1 + InverseSeries[Series[4*x/(1+x)-3*Log[1+x], {x, 0, 20}], x],x]* Range[0, 20]! (* _Vaclav Kotesovec_, Dec 26 2013 *)
%o A234293 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+A^4*intformal(1/(A^3+x*O(x^n)))); n!*polcoeff(A, n)}
%o A234293 for(n=0, 25, print1(a(n), ", "))
%o A234293 (PARI) {a(n)=local(A=1, X=x+x^2*O(x^n)); A=1+serreverse(4*X/(1+X)-3*log(1+X)); n!*polcoeff(A, n)}
%o A234293 for(n=0, 25, print1(a(n), ", "))
%Y A234293 Cf. A234292, A234294, A234295.
%K A234293 nonn
%O A234293 0,3
%A A234293 _Paul D. Hanna_, Dec 25 2013