(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger)
Written: April 19, 2013.
This is a follow-up of a fascinating conjecture made by Melkmau Zeleke in a talk given at the Rutgers Experimental Mathematics Seminar on April 18, 2013.
Added April 29, 2013: Here are interesting comments by Ira Gessel:
"There is a simple combinatorial interpretation of the rational functions F_m(z) in Shalosh B. Ekhad's paper "On a conjecture of Melkamu Zeleke". It counts Dyck paths with height at most m-1 (or to be more precise, "Dyck paths" that start at height 0 and end at height 1 that never go above height m-1, but these are almost the same as the genuine Dyck paths that return to height 0. Various related formulas can be found in my paper with Guoce Xin, to get your formula, set i=0, j=1, and k=m-1 in the second (unnumbered) formula at the top of page 11. Unfortunately our C is your zC2."