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

The Catalan-like numbers cn,0, defined by cn+1,k = rk−1cn,k−1 + skcn,k + tk+1cn,k+1 for n, k ≥ 0, c0,0 = 1, c0,k = 0 for k 6= 0, unify a substantial amount of well-known counting coefficients. Using an algebraic approach, Zhu showed that the sequence (cn,0)n≥0 is log-convex if rktk+1 ≤ sksk+1 for all k ≥ 0. Here we give a combinatorial proof of this result from the point of view of weighted Motzkin paths.