Geodesic
A combinatorial proof of the log-convexity of Catalan-like numbers
Journal of integer sequences, Tome 17 (2014) no. 5
The Catalan-like numbers $c_{n,0}$, defined by $\begin{align*}\{n+1,k}=r_{k-1}c_{n,k-1}+s_kc_{n,k}+t_{k+1}c_{n,k+1}{ for n,kgeq0 },\\ \{0,0}=1, c_{0,k}=0 { for kneq0 }, \end{align*}$ 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.
Classification : 05A19, 05A20
Keywords: Catalan-like number, log-convexity, weighted Motzkin path 
@article{JIS_2014__17_5_a3,
     author = {Sun,  Hua and Wang,  Yi},
     title = {A combinatorial proof of the log-convexity of {Catalan-like} numbers},
     journal = {Journal of integer sequences},
     year = {2014},
     volume = {17},
     number = {5},
     zbl = {1287.05013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2014__17_5_a3/}
}
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Sun,  Hua; Wang,  Yi. A combinatorial proof of the log-convexity of Catalan-like numbers. Journal of integer sequences, Tome 17 (2014) no. 5. http://geodesic.mathdoc.fr/item/JIS_2014__17_5_a3/