Skew Dyck Paths Having no Peaks at Level 1

Skew Dyck paths are a variation of Dyck paths, where in addition to the steps (1, 1) and (1, -1), a south-west step (-1, -1) is also allowed, provided that the path does not intersect itself. Replacing the south-west step by a red south-east step, we end up with decorated Dyck paths. Sequence A128723 of the On-Line Encyclopedia of Integer Sequences (OEIS) considers such paths where peaks at level 1 are forbidden. We provide a thorough analysis of a more general scenario, namely partial decorated Dyck paths, ending on a prescribed level j, both from left-to-right and from right-to-left (decorated Dyck paths are not symmetric). The approach is completely based on generating functions.