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%I #20 Nov 02 2023 10:39:50
%S 1,1,2,7,29,129,602,2910,14447,73234,377487,1972568,10425930,55640282,
%T 299403552,1622701202,8850030065,48534971244,267486182192,
%U 1480673755443,8228819436898,45895682480965,256815165790211,1441321638029496,8111194646903282
%N G.f. satisfies A(x) = 1 + x*A(x)^3 - x^2*A(x)^2.
%C Compare to the g.f. C(x) for the Catalan numbers (A000108): C(x) = 1 + x*C(x)^3 - x^2*C(x)^4 = 1 + x*C(x)^2.
%F Recurrence: 2*n*(2*n+1)*(244*n^3 - 1713*n^2 + 3767*n - 2550)*a(n) = 3*(2196*n^5 - 17613*n^4 + 49628*n^3 - 59841*n^2 + 30478*n - 5184)*a(n-1) - 18*(244*n^5 - 2323*n^4 + 8013*n^3 - 12252*n^2 + 7774*n - 1260)*a(n-2) - (n-4)*(244*n^4 - 1713*n^3 + 4172*n^2 - 3333*n - 378)*a(n-3) - 36*(n-5)*(n-3)*(5*n + 2)*a(n-4) - 4*(n-6)*(n-4)*(244*n^3 - 981*n^2 + 1073*n - 252)*a(n-5). - _Vaclav Kotesovec_, Sep 10 2013
%F a(n) ~ c*d^n/n^(3/2), where d = 5.991151107674316485... is the root of the equation -4 - 4*d - 5*d^2 - 23*d^3 + 4*d^4 = 0 and c = 0.214566307956522153666714736272121899143... - _Vaclav Kotesovec_, Sep 10 2013
%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-4*k,k) * binomial(3*n-5*k,n-2*k) / (2*n-3*k+1). - _Seiichi Manyama_, Nov 02 2023
%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 29*x^4 + 129*x^5 + 602*x^6 +...
%e Related expansions:
%e A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 76*x^4 + 344*x^5 + 1627*x^6 +...
%e A(x)^3 = 1 + 3*x + 9*x^2 + 34*x^3 + 147*x^4 + 678*x^5 + 3254*x^6 +...
%e where a(2) = 3 - 1; a(3) = 9 - 2; a(4) = 34 - 5; a(5) = 147 - 18; ...
%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 - 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-x^2*A^2+x*O(x^n));polcoeff(A,n);
%o (PARI) a(n) = sum(k=0, n\2, (-1)^k*binomial(3*n-4*k, k)*binomial(3*n-5*k, n-2*k)/(2*n-3*k+1)); \\ _Seiichi Manyama_, Nov 02 2023
%Y Cf. A104979, A200754.
%Y Cf. A000108, A137265, A200753.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 21 2011
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