The sequence [A006318](https://oeis.org/A006318) at OEIS stands for the [Schröder numbers](https://en.wikipedia.org/wiki/Schr%C3%B6der_number). 



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> 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).
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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](https://hrcak.srce.hr/19) 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.



Why is the number of Perfect Matchings in a triangular grid equivalent to the number of Royal Paths?
I was able to find the result at **Ex. 6.39 s** in Stanley's EC Vol.2. The reference given there is to



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> Ciucu, M. "[Perfect Matchings of Cellular Graphs](https://doi.org/10.1023/A:1022408900061)" *Journal of Algebraic Combinatorics* **5** (1996), 87-103
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In theorem 4.1, Ciucu computes the generating function of the perfect matchings in a triangular grid of squares and concludes that it is the same as the generating function of Schröder numbers. In the remark that follows the proof of theorem 4.1, a bijection between matchings and paths is also given.


