The height of two types of generalised Motzkin paths

Consider Motzkin paths which are lattice paths in the plane starting at the origin, running weakly above the x-axis and after n unit steps returning at the point ( n , 0 ) . The allowed steps are the up and down steps ( 1 , 1 ) and ( 1 , − 1 ) respectively and certain horizontal steps. We consider two types of horizontal steps that have attracted recent attention in the literature. First, we consider unit horizontal steps ( 1 , 0 ) coloured with k colours, secondly, we look at paths where the horizontal steps are of length k, for a non-negative integer k. Using generating functions, we study the sum of heights of such paths of size n. With the use of the Mellin transform, we find asymptotic expressions for the mean heights as n tends to infinity.