On a Conjecture of Melkamu Zeleke Shalosh B. EKHAD

In his fascinating talk[Z], Melkamu Zeleke made the following conjecture. 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} ,

then for all positive integers m, the formal power series

F_{m}(z) := \frac{zC^{2} − (zC^{2})^{m}}{1 − (zC^{2})^{m+1}} ,

are all 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 rigorously proved by only checking finitely many special cases. It turns out that checking it for 2 ≤ m ≤ 10 suffices! (why?). Just to play it safe I checked the proposition below for 2 ≤ m ≤ 100. Proposition Let N_{m}(z) be the coefficient of X^{m} in the Maclaurin expansion of

\frac{−z − zX + z^{2}X^{2}}{1 + (1 − 2 z) X^{2} + z^{2}X^{4}} ,

and Let D_{m}(z) be the coefficient of X^{m} in the Maclaurin expansion of

\frac{−1 + z + (2 z − 1) X − z^{2}X^{2} − z^{2}X^{3}}{1 + (1 − 2 z) X^{2} + z^{2}X^{4}} ,

then

F_{m}(z) = \frac{N_{m−2}(z)}{D_{m−2}(z)} .

Reference

[Z] Melkamu Zeleke, 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)

Shalosh B. Ekhad, Mathematics Department, Rutgers University (New Brunswick), Piscataway,

NJ 08854, USA. c/o zeilberg at math dot rutgers dot edu

April 19, 2013