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.