Motzkin paths with two variants of level steps on odd levels -- a kernel method approach

The sequence A176677 in the Encyclopedia of Integer Sequences enumerates Motzkin paths where two types of horizontal steps may occur, but only on odd indexed levels. We show how to perform the enumeration, also dealing with partial such Motzkin paths leading to a particular level or to any level (open paths). The method is the kernel method where functional equations are manipulated in a suitable way. The coefficients of sequence A176677 satisfy a holonomic recursion that was recently discussed on the arxiv. We show how this can be established in an (almost) automatic fashion. Eventually we switch the roles of odd' and even’. One could also allow more versions of horizontal steps but we leave this to the interested readers.