On enumeration formulas for weighted Motzkin paths

We give a combinatorial proof of an enumeration formula for weighted Motzkin paths, which was first proved by Corteel, Josuat-Vergès, Prellberg, and Rubey, and is related to various objects such as PASEP, matrix ansatz, permutations, permutation tableaux, and moments of Laguerre polynomials. We then generalize our method to give a formula for weighted Motzkin paths with two additional parameters which were studied by Josuat-Vergès recently.