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%I #31 Apr 29 2015 14:49:08
%S 1,1,2,6,20,72,273,1073,4333,17869,74937,318601,1370113,5949201,
%T 26046727,114857599,509669295,2274146599,10197234215,45925646367,
%U 207656685443,942302814363,4289903653615,19588180438263,89685571667763
%N G.f. satisfies: A(x) = 1 + x*A(x) + x^2*A(x)^4.
%C Compare to a g.f. C(x) of Catalan numbers: C(x) = 1 + x*C(x) + x^2*C(x)^3.
%H Robert Israel, <a href="/A186996/b186996.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: A(x) = (1/x)*Series_Reversion( 2*x^3/(1 - sqrt(1-4*x^2-4*x^3)) ).
%F G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x*A(x)^3) / (1 + x*A(x)^2).
%F Recurrence: 3*(n-2)*n*(2*n-5)*(3*n-2)*(3*n+2)*a(n) = 3*(2*n-5)*(2*n-1)*(18*n^3 - 54*n^2 + 36*n - 5)*a(n-1) + (2*n-3)*(94*n^4 - 564*n^3 + 1222*n^2 - 1128*n + 331)*a(n-2) + 3*(2*n - 5)*(2*n - 1)*(18*n^3 - 108*n^2 + 198*n - 103)*a(n-3) - 3*(n-3)*(n-1)*(2*n-1)*(3*n-11)*(3*n-7)*a(n-4). - _Vaclav Kotesovec_, Sep 10 2013
%F a(n) ~ c*d^n/n^(3/2), where d = (9 + 8*sqrt(3) + 4*sqrt(12 + 9*sqrt(3)))/9 = 4.874032512954972962... is the root of the equation 27 - 108*d - 94*d^2 - 108*d^3 + 27*d^4 = 0 and c = sqrt(11/(8*(-44 + sqrt(748 + 891*sqrt(3)))*Pi)) = 0.336422381089368230542882135982348331566666028... - _Vaclav Kotesovec_, Sep 10 2013, updated Apr 27 2015
%F a(n) = sum(k=0..n, binomial(3n-2k+1, k)*binomial(k,n-k)*1/(3n-2k+1)). - _Michael D. Weiner_, Apr 23 2015
%F G.f.: A(x) = hypergeom([1/4, 1/2, 3/4], [2/3, 4/3], (256/27)*x^2/(x-1)^4)/(1-x). - _Robert Israel_, Apr 28 2015
%e G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 72*x^5 + 273*x^6 +...
%e Related expansions:
%e A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 56*x^4 + 208*x^5 + 806*x^6 +...
%e A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 114*x^4 + 438*x^5 + 1739*x^6 +...
%e A(x)^4 = 1 + 4*x + 14*x^2 + 52*x^3 + 201*x^4 + 800*x^5 + 3260*x^6 +...
%p S:= series(RootOf(y - 1 - x*y - x^2*y^4, y), x, 101):
%p seq(coeff(S,x,i),i=0..100); # _Robert Israel_, Apr 23 2015
%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 + x^2*AGF^4 - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* _Vaclav Kotesovec_, Sep 10 2013 *)
%t Table[Sum[Binomial[3n-2k+1,k]*Binomial[k,n-k]/(3n-2k+1),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Apr 27 2015 after _Michael D. Weiner_ *)
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*A+x^2*(A+x*O(x^n))^4);polcoeff(A,n)}
%o (PARI) {a(n)=polcoeff((1/x)*serreverse(2*x^3/(1 - sqrt(1-4*x^2-4*x^3 +x^3*O(x^n)))),n)}
%Y Cf. A182454.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 01 2011
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