Some Properties and Combinatorial Implications of Weighted Small Schr\"oder Numbers

The $n^{\text{th}}$ small Schr\"oder number is $s(n) = \sum_{k \geq 0} s(n,k)$, where $s(n,k)$ denotes the number of plane rooted trees with $n$ leaves and $k$ internal nodes that each has at least two children. In this manuscript, we focus on the weighted small Schr\"oder numbers $s_d(n) = \sum_{k \geq 0} s(n,k) d^k$, where $d$ is an arbitrary fixed real number. We provide recursive and asymptotic formulas for $s_d(n)$, as well as some identities and combinatorial interpretations for these numbers. We also establish connections between $s_d(n)$ and several families of Dyck paths.