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A258969 E.g.f.: A'(x) = 1 + A(x)^3, with A(0)=1.

%I A258969 #21 Jun 06 2024 12:25:52
%S A258969 1,2,6,42,390,4698,69174,1203498,24163110,549811962,13982486166,
%T A258969 393026414922,12099527531910,404881353252378,14632253175107574,
%U A258969 567974815524008298,23567351945550373350,1040985881615266375482,48767788927611416600406,2415210691383917131432842
%N A258969 E.g.f.: A'(x) = 1 + A(x)^3, with A(0)=1.
%C A258969 Conjecture: A227250(n+1) = a(n).
%H A258969 Vaclav Kotesovec, <a href="/A258969/b258969.txt">Table of n, a(n) for n = 0..200</a>
%F A258969 a(n) ~ (3/(Pi/sqrt(3)-log(2)))^(n+1/2) * n^n / exp(n).
%F A258969 E.g.f.: 1 + Series_Reversion( Integral 1/((2+x)*(1+x+x^2)) dx ). - _Paul D. Hanna_, Jun 16 2015
%e A258969 A(x) = 1 + 2*x + 6*x^2/2! + 42*x^3/3! + 390*x^4/4! + 4698*x^5/5! + ...
%e A258969 A'(x) = 2 + 6*x + 21*x^2 + 65*x^3 + 783*x^4/4 + 11529*x^5/20 + ...
%e A258969 1 + A(x)^3 = 2 + 6*x + 21*x^2 + 65*x^3 + 783*x^4/4 + 11529*x^5/20 + ...
%t A258969 nmax=20; Subscript[a,0]=1; egf=Sum[Subscript[a,k]*x^k, {k,0,nmax+1}]; Table[Subscript[a,k]*k!, {k,0,nmax}] /.Solve[Take[CoefficientList[Expand[1+egf^3-D[egf,x]],x],nmax]==ConstantArray[0,nmax]][[1]]
%o A258969 (PARI) {a(n) = local(A=1); A = 1 + serreverse( intformal( 1/((2+x)*(1+x+x^2) +x*O(x^n)) )); n!*polcoeff(A,n)}
%o A258969 for(n=0,25,print1(a(n),", ")) \\ _Paul D. Hanna_, Jun 16 2015
%Y A258969 Cf. A000831, A024396, A193534, A227250, A258970, A258971, A258880, A258994.
%K A258969 nonn
%O A258969 0,2
%A A258969 _Vaclav Kotesovec_, Jun 15 2015