The Catalan-like numbers <I>c</I><SUB><I>n</I>,0</SUB>, defined by
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<IMG
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 SRC="img1.gif"
 ALT="\begin{align*}&c_{n+1,k}=r_{k-1}c_{n,k-1}+s_kc_{n,k}+t_{k+1}c_{n,k+1}\text{ for $n,k\geq 0$ },\\
&c_{0,0}=1, c_{0,k}=0 \text{ for $k\neq 0$ },
\end{align*}">
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unify a substantial amount of well-known counting coefficients.
Using an algebraic approach,
Zhu showed that the sequence 
<!-- MATH: $(c_{n,0})_{n\geq 0}$ -->
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is log-convex
if 
<!-- MATH: $r_{k}t_{k+1}\leq s_{k}s_{k+1}$ -->
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 ALT="$r_{k}t_{k+1}\leq s_{k}s_{k+1}$">
for all <IMG
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 ALT="$k\geq 0$">.
Here we give a combinatorial proof of this result
from the point of view of weighted Motzkin paths.