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%I #13 Sep 11 2013 08:35:57
%S 1,1,5,23,123,714,4371,27789,181703,1214166,8254642,56915989,
%T 397055129,2797402939,19875881883,142255845279,1024672816559,
%U 7422364394376,54034027247172,395121014784661,2900916414217452,21375452831552800,158025136138734047,1171770189246765570
%N G.f. satisfies: A(x) = 1 + x*A(x)^3 + 2*x^2*A(x)^2 + x^3*A(x).
%H Vincenzo Librandi, <a href="/A200028/b200028.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. satisfies: A(x) = 1 + x*A(x)*(A(x) + x)^2.
%F Recurrence: 2*n*(2*n+1)*(269*n^4 - 2870*n^3 + 11215*n^2 - 17566*n + 7440)*a(n) = 3*(2421*n^6 - 28251*n^5 + 127807*n^4 - 266413*n^3 + 251420*n^2 - 110096*n + 18072)*a(n-1) + 18*(538*n^6 - 7085*n^5 + 36497*n^4 - 91805*n^3 + 108177*n^2 - 38330*n - 12672)*a(n-2) + 2*(n-4)*(1076*n^5 - 11480*n^4 + 46507*n^3 - 83710*n^2 + 43773*n + 29106)*a(n-3) + 36*(n-5)*(2*n-3)*(7*n^2 - 3*n + 38)*a(n-4) - 4*(n-6)*(269*n^5 - 3139*n^4 + 13659*n^3 - 25097*n^2 + 16486*n - 4230)*a(n-5) - 2*(n-7)*(2*n-11)*(269*n^4 - 1794*n^3 + 4219*n^2 - 2670*n - 1512)*a(n-6). - _Vaclav Kotesovec_, Sep 10 2013
%F a(n) ~ c*d^n/n^(3/2), where d = 7.918229914943462873... is the root of the equation 4 + 4*d - 8*d^3 - 36*d^4 - 27*d^5 + 4*d^6 = 0 and c = 0.283739249... - _Vaclav Kotesovec_, Sep 10 2013
%e G.f.: A(x) = 1 + x + 5*x^2 + 23*x^3 + 123*x^4 + 714*x^5 + 4371*x^6 +...
%e Related expansions:
%e A(x)^2 = 1 + 2*x + 11*x^2 + 56*x^3 + 317*x^4 + 1904*x^5 + 11929*x^6 +...
%e A(x)^3 = 1 + 3*x + 18*x^2 + 100*x^3 + 597*x^4 + 3714*x^5 + 23858*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*(AGF+x)^2-AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* _Vaclav Kotesovec_, Sep 10 2013 *)
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*A^3+2*x^2*A^2+x^3*A^1+x*O(x^n));polcoeff(A,n)}
%Y Cf. A200029, A200030.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 12 2011
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