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In this paper, we perform an arithmetization of well-formed parenthesisstrings with zeros (Motzkin words) and of corresponding Motzkin paths. Thetransformations used are reminiscent of G\"odel numbering for mathematicalobjects of some formal language. We construct a Motzkin series that is as closeas possible to natural numbers by many formal features. Parenthesis strings areencoded by ternary codes and corresponding natural numbers, Motzkin numbers ofthe 2nd kind, which made it possible to formalize and simplify the analysis ofthe successor function, to specify and clarify the procedure for selecting asuccessor. In the process of arithmetization of well-wormed parenthesisstrings, various special numbers appeared. In this regard, in conclusion wewill talk a little about numbers of the form $3^n+2$ and mirror numbers$2*3^n+1$.

Arithmetization of well-formed parenthesis strings. Motzkin Numbers of the Second Kind