Combinatorial Proofs of Addition Formulas
Keywords:
Lattice paths, Catalan numbers, Motzkin numbers, Large Schr\"{o}der numbers, Hankel determinant
Abstract
In this paper we give a combinatorial proof of an addition formula for weighted partial Motzkin paths. The addition formula allows us to determine the $LDU$ decomposition of a Hankel matrix of the polynomial sequence defined by weighted partial Motzkin paths. As a direct consequence, we get the determinant of the Hankel matrix of certain combinatorial sequences. In addition, we obtain an addition formula for weighted large Schröder paths.