Journal of Integer Sequences, Vol. 17 (2014), Article 14.5.2

A Combinatorial Proof of the Log-Convexity of Catalan-Like Numbers

Hua Sun and Yi Wang
School of Mathematical Sciences
Dalian University of Technology
Dalian 116024
PR China


The Catalan-like numbers cn,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}\text{ for $n,k\geq 0$ },\\
&c_{0,0}=1, c_{0,k}=0 \text{ for $k\neq 0$ },
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.

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(Concerned with sequences A000108 A000110 A000957 A000984 A001006 A002212 A005043 and A006318.)

Received January 28 2014; revised version received March 17 2014. Published in Journal of Integer Sequences, March 23 2014.

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