On Generalizing Motzkin Numbers Using $k$-Trees

Abstract

We use $k$-trees to generalize the sequence of Motzkin numbers and show that Baxter’s generalization of Temperley-Lieb operators is a special case of our generalization of Motzkin numbers. We also obtain a recursive summation formula for the terms of $3$-Motzkin numbers and investigate some asymptotic properties of the terms of $k$-Motzkin numbers.