Peakless Motzkin paths of bounded height

There was recent interest in Motzkin paths without peaks (peak: up-step followed immediately by down-step); additional results about this interesting family is worked out. The new results are the enumeration of such paths that live in a strip $[0..\ell]$, and as consequence the asymptotics of the average height, which is given by $2\cdot 5^{-1/4}\sqrt{\pi n}$. Methods include the kernel method and singularity analysis of generating functions.