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

1, 0, 3, 0, 12, 0, 54, 0, 264, 0, 1365, 0, 7344, 0, 40698, 0, 230736, 0, 1332045, 0, 7803900, 0, 46280520, 0, 277294752, 0, 1676056044, 0, 10207492512, 0, 62576750610, 0, 385856498592, 0, 2391506423565, 0, 14890534347780, 0

Offset

3,3

Links

Table of n, a(n) for n=3..40.

Formula

a(2n)=0; a(2n+1) = 6*C(3*n-3,n-2)/n for n>1 with a(1)=0, a(3)=1.

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.

Define A_{n+1}(x) = A( A_{n}(x) ) with A_0(x)=x, then

A_{n}(x) = (x*G(x^2))^(3^n) - (x*G(x^2))^(3^(n+1)) and so

Sum_{n>=0} A_{n}(x) = x*G(x^2) where G(x) = g.f. of A001764.

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

Example

A(x) = x^3 + 3*x^5 + 12*x^7 + 54*x^9 + 264*x^11 + 1365*x^13 + ...
x^3 - x^9 = (x-x^3)^3 + 3*(x-x^3)^5 + 12*(x-x^3)^7 + 54*(x-x^3)^9 + ...
Let G(x) = 1 + x*G(x)^3 = g.f. of A001764, then
A(x) = (x*G(x^2))^3 - (x*G(x^2))^9;
A(A(x)) = (x*G(x^2))^9 - (x*G(x^2))^27;
A(A(A(x))) = (x*G(x^2))^27 - (x*G(x^2))^81; etc.
Related expansions:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + ...
A(A(x)) = x^9 + 9*x^11 + 63*x^13 + 408*x^15 + 2565*x^17 + ...
A(A(A(x))) = x^27 + 27*x^29 + 432*x^31 + 5355*x^33 + 56943*x^35 + ...

Prog

(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)))

Crossrefs

Cf A001764 (enumerates ternary trees).

Sequence in context: A019264 A028851 A132221 * A194093 A055314 A110890

Adjacent sequences: A140090 A140091 A140092 * A140094 A140095 A140096

Keyword

nonn

Author

Paul D. Hanna, May 11 2008

Status

approved