The connection between weighted counts of Motzkin paths and moments of orthogonal polynomials is well known. We look at the inverse generating function of Motzkin paths with weighted horizontal steps, and relate it to Chebyshev polynomials of the second kind. The inverse can be used to express the number of paths ending at a certain height in terms of those ending at height 0. Paths of a more general horizontal step length w are also investigated. We suggest three applications for the inverse. First, we use the inverse Motzkin matrix to express some Hankel determinants of Motzkin paths. Next, we count the paths inside a horizontal band using a ratio of inverses. Finally, for Schr¨oder paths (w = 2) we write the number of paths inside the same band that end on the top side of the band in terms of those ending at height 0, with the help of the inverse Schr¨oder matrix.
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