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Dyck paths (also balanced brackets and Dyck words) are among the most heavilystudied Catalan families. This paper is a continuation of [2, 3, 4]. In thepaper we are dealing with the numbering of Dyck paths, with the resultingnumbers, the terms of the OEIS sequence A036991, which encode Dyck paths andwhich we have called the Dyck numbers. In this sequence, it turned out to bepromising to investigate nested term patterns. The sequence is constructedentirely from nested patterns, with each pattern having an infinite number ofcopies. We have already considered repeated triplets of adjacent odd numbers;copies of such triplets in the process of expansion form quite complexstructures, the analysis of which allows us to understand the structure ofA036991. Additionally, we note that the composition of the terms of eachpattern is repeated many times by a simple shift (offset) of Dyck numbers bysome fixed value. As a result, pattern copies give us important parts of mostlevels and even give us new levels.

Dyck Numbers, IV. Nested patterns in OEIS A036991