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In the present paper we realize a simple idea: express the coefficients of the series ${{g}^{m}}\\left( x \\right)$ in terms of the coefficients of the series $B\\left( x \\right)$. Obtained expansion has a bright combinatorial character, sheds light on the connection of the pseudo-involution in "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1707.00900","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-07-04T10:39:03Z","cross_cats_sorted":[],"title_canon_sha256":"5e8ec28dec4442ff18eed97df1c901a0bf0183caa1566048e3b41b4adb1f2755","abstract_canon_sha256":"406693b2172391be6c8674ae4e3ec77e2d00a4804177ad2fe1c17e75b9753af3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:56.700510Z","signature_b64":"qMw65ejbRB0Xkfv+YnqfLtBK91iaRg0irJXxpHqucZDSvXk0AACKBzLpsN1O3kkWpg5/0MOWcERSiF/VcpIZDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f38078265017d438e228e2b9b47ba46e1f57cef791fcaf8bb7a0290a782f269c","last_reissued_at":"2026-05-18T00:40:56.699861Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:56.699861Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$B$-expansion of pseudo-involution in the Riordan group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"E. 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