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A124214 E.g.f.: exp(x) / (2 - exp(3*x))^(1/3).

%I A124214 #24 Sep 01 2016 13:02:31
%S A124214 1,2,10,98,1402,26162,601930,16462658,521659162,18791451602,
%T A124214 758345497450,33889063202018,1661229537252922,88627461127536242,
%U A124214 5112116659677605770,317007674364657538178,21030558126242472270682
%N A124214 E.g.f.: exp(x) / (2 - exp(3*x))^(1/3).
%H A124214 G. C. Greubel, <a href="/A124214/b124214.txt">Table of n, a(n) for n = 0..350</a>
%F A124214 a(n) ~ Gamma(2/3)*3^(n+1/2)*n^(n-1/6)/(sqrt(2*Pi)*exp(n)*(log(2))^(n+1/3)). - _Vaclav Kotesovec_, Jun 26 2013
%F A124214 E.g.f. A(x) satisfies: A'(x) = A(x) + A(x)^4. - _Paul D. Hanna_, Dec 18 2013
%F A124214 E.g.f. A(x) satisfies: A(x) = exp(x + Integral A(x)^3 dx) with A(0)=1. - _Paul D. Hanna_, Dec 18 2013
%F A124214 a(n) = 2^(-1/3) * Sum_{k >= 0} (1/18)^k*A004987(k)*(3*k + 1)^n = 2^(-1/3) * Sum_{k >= 0} (-1/2)^k*binomial(-1/3, k)*(3*k + 1)^n. Cf. A124212 and A229558. - _Peter Bala_, Aug 30 2016
%p A124214 A124214 := proc(n)
%p A124214     exp(x)/root[3](2-exp(3*x)) ;
%p A124214     coeftayl(%,x=0,n)*n! ;
%p A124214 end proc: # _R. J. Mathar_, Dec 19 2013
%t A124214 CoefficientList[Series[Exp[x]/(2-Exp[3*x])^(1/3), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 26 2013 *)
%o A124214 (PARI) a(n)=local(A=1+x); for(i=1, n, A=1+intformal(A+A^4+x*O(x^n))); n!*polcoeff(A, n)
%o A124214 for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Dec 18 2013
%Y A124214 Cf. A229558, A004987, A124212.
%K A124214 nonn,easy
%O A124214 0,2
%A A124214 _Karol A. Penson_, Oct 19 2006