The $l$-th Catalan number ${2l\choose l}\frac{1}{l+1}$ is equal
to the number of sequences $s\_0,\ldots,s\_{l+1}$ of length $l+2$ with the following
properties:


(1) $s\_0=s\_{l+1}=1$ and $s\_1,\ldots,s\_l$ are integers $\geq 2$,


(2) $s\_i$ divides $s\_{i-1}+s\_i+s\_{i+1}$ for $i=1,\ldots,l$.


(The proof is easy: consecutive terms $s\_is\_{i+1}$ in such sequences encode leaves of finite binary trees with $l+1$ leaves. This encoding of binary trees has nice properties: the integer $(s\_{i-1}+s\_i+s\_{i+1})/s\_i$ is the distance between the consecutive leaves
encoded by $s\_{i-1},s\_i$ and $s\_i,s\_{i+1}$. Leaves correspond always to coprime integers $s\_i,s\_{i+1}$ and heights of leaves can be computed by applying Euclid's algorithm (essentially by summing all quotients occuring during the computation) to the coprime pair
$(s\_i,s\_{i+1})$. Interior vertices of the binary tree encoded by $s\_0,\ldots,s\_{l+1}$ correspond to segments $s\_i,s\_{i+1},\ldots,s\_{i+k}$ such that $s\_{i+1},\ldots,s\_{i+k-1}$
are all strictly larger than $\max(s\_i,s\_{i+k})$ (and the height of such an interior vertex is given as for leaves by Euclid's algorithm applied to $s\_i,s\_{i+k}$ which are always coprime). In particular the
binary tree of the left child of the root vertex (for $l\geq 1$) is
encoded by the initial terms $s\_0,\ldots,s\_i=2$. The subtree for the right child corresponds to $s\_i=2,\ldots,s\_{l+1}$.)


Examples (with integers concatenated):


$l=0$: $11$,


$l=1$: $121$,


$l=2$: $1231,1321$,


$l=3$: $12341,12531,13231,13521,14321$.


(Recipe for recursive constructions: All such sequences of length $l+3$ are obtained from such sequences $l+2$ by inserting in all possible ways between two adjacent terms their sum.)


*Is there a reference for this description?* (I could not find it in Stanley's list but a mistake on my behalf is likely.)



Reference for a definition of Catalan numbers
I'm converting my comments to an answer. This interpretation of the Catalan numbers is indeed well-known. It appears for instance as Exercise 92 in Chapter 2 of Stanley's book on [Catalan numbers](https://doi.org/10.1017/CBO9781139871495), as well as [Exercise 6.19(iii)](https://math.mit.edu/~rstan/ec/catalan.pdf) in his *Enumerative Combinatorics, Vol. 2*. It is also discussed, for instance, in the paper [Sum-Difference Sequences and Catalan Numbers](https://doi.org/10.1007/s006050050025) by Aigner & Schulze, who attribute this interpretation to Van Lint.


(By the way, it is a little unclear why you write "$s\_i$ divides $s\_{i-1}+s\_i+s\_{i+1}$" instead of "$s\_i$ divides $s\_{i-1}+s\_{i+1}$", but of course that amounts to the same thing.)


