Summary: The Catalan-like numbers $c_{n,0}$, defined by $\begin{align*}\&c_{n+1,k}=r_{k-1}c_{n,k-1}+s_kc_{n,k}+t_{k+1}c_{n,k+1}{ for n,kgeq0 },\\ \&c_{0,0}=1, c_{0,k}=0 { for kneq0 }, \end{align*}$ unify a substantial amount of well-known counting coefficients. Using an algebraic approach, Zhu showed that the sequence $(c_{n,0})_{n\geq 0}$ is log-convex if $r_{k}t_{k+1}\leq s_{k}s_{k+1}$ for all $k\geq 0$. Here we give a combinatorial proof of this result from the point of view of weighted Motzkin paths.

05A19, 05A20

Catalan-like number, log-convexity, weighted Motzkin path