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{
\bf 
On a  Conjecture of Melkamu Zeleke
}
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{
\it Shalosh B. EKHAD
}
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In his fascinating talk[Z], Melkamu Zeleke made the following conjecture.

{\bf Conjecture}: Let $C(z)$ be the generating function of the Catalan numbers, i.e. the formal power series
satisfying
$$
C(z)=1+zC(z)^2 \quad ,
$$
then for all positive integers  $m$, the formal  power series
$$
F_m(z):=\frac{zC^2-(zC^2)^m}{1-(zC^2)^{m+1}} \quad ,
$$
are all {\bf rational functions} of $z$.


In this short note we will give an explicit description of $F_m(z)$ as a rational function, that, once conjectured,
can be {\bf rigorously} proved by {\bf only} checking finitely many special cases. 
It turns out that checking it for $2 \leq m \leq 10$ suffices! (why?).
Just to play it safe I checked the proposition below for $2 \leq m \leq 100$.


{\bf Proposition} Let $N_m(z)$ be the coefficient of $X^m$ in the  Maclaurin expansion of
$$
{\frac {-z-zX+{z}^{2}{X}^{2}}{1+ \left( 1-2\,z \right) {X}^{2}+{z}^{2}{X}^{4}}} \quad,
$$
and Let $D_m(z)$ be the coefficient of $X^m$ in the  Maclaurin expansion of
$$
{\frac {-1+z+ \left( 2\,z -1 \right) X- {z}^{2} {X}^{2}- {z}^{2} {X}^{3}}{1+ \left( 1 - 2\,z \right) {X}^{2}+{z}^{2}{X}^{4}}} \quad,
$$
then
$$
F_m(z)=\frac{N_{m-2}(z)}{D_{m-2}(z)} \quad . \quad \halmos
$$

{\bf Reference}

[Z] Melkamu Zeleke, {\it On Subsets of Ordered Trees Enumerated by a Subsequence of Fibonacci Numbers },
talk delivered at the Rutgers Experimental Mathematics Seminar, April 18, 2013.

(Based  on joint work with Mahendra Jani)


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Shalosh B. Ekhad, Mathematics Department, Rutgers University (New Brunswick), Piscataway, NJ 08854, USA.
c/o {\tt zeilberg at math dot rutgers dot edu}

{\bf April 19, 2013}


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