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%I #9 Nov 24 2022 18:28:20
%S 1,0,3,0,12,0,54,0,264,0,1365,0,7344,0,40698,0,230736,0,1332045,0,
%T 7803900,0,46280520,0,277294752,0,1676056044,0,10207492512,0,
%U 62576750610,0,385856498592,0,2391506423565,0,14890534347780,0
%N G.f. satisfies: A(x - x^3) = x^3 - x^9.
%F a(2n)=0; a(2n+1) = 6*C(3*n-3,n-2)/n for n>1 with a(1)=0, a(3)=1.
%F G.f.: A(x) = x^3 + 3*x^5*G(x^2)^4 where G(x) = 1 + x*G(x)^3 = g.f. of A001764.
%F Define A_{n+1}(x) = A( A_{n}(x) ) with A_0(x)=x, then
%F A_{n}(x) = (x*G(x^2))^(3^n) - (x*G(x^2))^(3^(n+1)) and so
%F Sum_{n>=0} A_{n}(x) = x*G(x^2) where G(x) = g.f. of A001764.
%F D-finite with recurrence: 4*(n-2)*(n+2)*(n+1)*a(n)-3*(3*n-4)*(3*n-2)*n*a(n-2) = 0. - _Georg Fischer_, Nov 24 2022
%e A(x) = x^3 + 3*x^5 + 12*x^7 + 54*x^9 + 264*x^11 + 1365*x^13 + ...
%e x^3 - x^9 = (x-x^3)^3 + 3*(x-x^3)^5 + 12*(x-x^3)^7 + 54*(x-x^3)^9 + ...
%e Let G(x) = 1 + x*G(x)^3 = g.f. of A001764, then
%e A(x) = (x*G(x^2))^3 - (x*G(x^2))^9;
%e A(A(x)) = (x*G(x^2))^9 - (x*G(x^2))^27;
%e A(A(A(x))) = (x*G(x^2))^27 - (x*G(x^2))^81; etc.
%e Related expansions:
%e G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + ...
%e A(A(x)) = x^9 + 9*x^11 + 63*x^13 + 408*x^15 + 2565*x^17 + ...
%e A(A(A(x))) = x^27 + 27*x^29 + 432*x^31 + 5355*x^33 + 56943*x^35 + ...
%o (PARI) a(n)=if(n<3 || n%2==0,0,if(n==3,1,12*binomial(3*(n-1)/2-3,(n-1)/2-2)/(n-1)))
%Y Cf A001764 (enumerates ternary trees).
%K nonn
%O 3,3
%A _Paul D. Hanna_, May 11 2008
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