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A140093 G.f. satisfies: A(x - x^3) = x^3 - x^9. 0

%I

%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.

%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 are:

%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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Last modified September 18 14:52 EDT 2021. Contains 347527 sequences. (Running on oeis4.)