Bracket-and-bar Sequences

Let us define the set of regular bracket-and-bar sequences $R$ recursively. It is the set of strings that can be obtained following only the rules below:

• $\varepsilon \in R$ (empty string)
• $A, B \in R \Rightarrow AB \in R$ (concatenation)
• $A, B \in R \Rightarrow$($A$|$B$)$\in R$

For example, the sequences containing two triples "(|)" look as folows: "((|)|)", "(|(|))", "(|)(|)".

Establish a correspondence between regular bracket-and-bar sequences of certain length and integers, and implement that correspondence.