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%I #9 Sep 11 2013 09:14:38
%S 1,1,4,18,89,483,2765,16441,100553,628496,3996864,25778259,168216339,
%T 1108586737,7367790912,49326025289,332342006775,2251828394103,
%U 15333892329935,104883312564708,720280904233625,4964456716210287,34329831032740121,238109354053880367
%N G.f. satisfies: A(x) = 1 + x*A(x)^3 + x^2*A(x)^2 + x^3*A(x).
%F Recurrence: 2*n*(2*n+1)*(3887*n^4 - 54440*n^3 + 250111*n^2 - 457630*n + 282264)*a(n) = 3*(34983*n^6 - 524943*n^5 + 2755069*n^4 - 6525877*n^3 + 7248740*n^2 - 3486644*n + 563184)*a(n-1) + 9*(7774*n^6 - 128315*n^5 + 763466*n^4 - 2108189*n^3 + 2796852*n^2 - 1569116*n + 201744)*a(n-2) + (42757*n^6 - 769868*n^5 + 5072115*n^4 - 15417004*n^3 + 21905960*n^2 - 12286992*n + 1257984)*a(n-3) - 36*(11*n^2 - 51*n - 2)*(113*n^2 - 685*n + 924)*a(n-4) - 2*(27209*n^6 - 571543*n^5 + 4528416*n^4 - 16844723*n^3 + 29833927*n^2 - 22362054*n + 4746168)*a(n-5) - (n-7)*(38870*n^5 - 602705*n^4 + 3170125*n^3 - 6712006*n^2 + 5294904*n - 895104)*a(n-6) + 648*(n-8)*(3*n - 7)*(11*n^2 - 51*n - 2)*a(n-7) + 108*(n-9)*(n+4)*(11*n^2 - 51*n - 2)*a(n-8) + 3*(n - 10)*(n-7)*(3887*n^4 - 38892*n^3 + 110113*n^2 - 105180*n + 24192)*a(n-9). - _Vaclav Kotesovec_, Sep 11 2013
%F a(n) ~ c*d^n/n^(3/2), where d = 7.40643631692299994... is the root of the equation -3 + 10*d^3 + 14*d^4 - 11*d^6 - 18*d^7 - 27*d^8 + 4*d^9 = 0 and c = 0.268208461330470914429962064... - _Vaclav Kotesovec_, Sep 11 2013
%e G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 89*x^4 + 483*x^5 + 2765*x^6 +...
%e Related expansions:
%e A(x)^2 = 1 + 2*x + 9*x^2 + 44*x^3 + 230*x^4 + 1288*x^5 + 7532*x^6 +...
%e A(x)^3 = 1 + 3*x + 15*x^2 + 79*x^3 + 435*x^4 + 2517*x^5 + 15064*x^6 +...
%t nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[1 + x*AGF^3 + x^2*AGF^2 + x^3*AGF - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* _Vaclav Kotesovec_, Sep 11 2013 *)
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*A^3+x^2*A^2+x^3*A^1+x*O(x^n));polcoeff(A,n)}
%Y Cf. A200028, A200030.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 12 2011
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