They describes the number of lattice paths from the southwest corner $(0,0)$ of an $n\times n$ grid to the northeast corner $(n,n)$, using only single steps north, $(0,1)$, northeast $(1,1)$ or east $(1,0)$, that do not rise above the SW–NE diagonal (sometimes called Royal Paths).
Also, according to OEIS, they correspond to the number of perfect matchings in number of perfect matchings in a triangular grid of n squares (n = 1, 4, 9, 16, 25, ...).
Tomislav Doslic wrote a very nice paper on a similar relation between perfect matchings in a hexagonal grid and the corresponding lattice path. He also mentions the triangular case, and cites the book "Enumerative Combinatorics" vol2, by Stanley, Ex. 6.39 - where I cannot find an answere.
Therefore I very much hope for some hints or further literature references on the question given in the title.