{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:N425PND5S5QFMPHWVJBETAGLUQ","short_pith_number":"pith:N425PND5","schema_version":"1.0","canonical_sha256":"6f35d7b47d9760563cf6aa424980cba4245c71a49e6e426f47077774d6efe7b4","source":{"kind":"arxiv","id":"1608.05657","version":3},"attestation_state":"computed","paper":{"title":"Motzkin numbers and related sequences modulo powers of $2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian Krattenthaler (Universit\\\"at Wien), Thomas W. M\\\"uller (Queen Mary & Westfield College, University of London)","submitted_at":"2016-08-19T16:27:45Z","abstract_excerpt":"We show that the generating function $\\sum_{n\\ge0}M_n\\,z^n$ for Motzkin numbers $M_n$, when coefficients are reduced modulo a given power of $2$, can be expressed as a polynomial in the basic series $\\sum _{e\\ge0} ^{} {z^{4^e}}/( {1-z^{2\\cdot 4^e}})$ with coefficients being Laurent polynomials in $z$ and $1-z$. We use this result to determine $M_n$ modulo $8$ in terms of the binary digits of~$n$, thus improving, respectively complementing earlier results by Eu, Liu and Yeh [Europ. J. Combin. 29 (2008), 1449-1466] and by Rowland and Yassawi [J. Th\\'eorie Nombres Bordeaux 27 (2015), 245-288]. An"},"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":"1608.05657","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-19T16:27:45Z","cross_cats_sorted":[],"title_canon_sha256":"8952e50aebb534087bdfe4689f1e82fed3749c4ff62dae3ff7a575a5ba8d9558","abstract_canon_sha256":"0f231979f99a5b2fdc8ef22657504237a638db7d3177807bab517988d1a59386"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:35.701427Z","signature_b64":"kvayp1ZsVy6C++lLmOn8+qrzQlIHPE4du82BOJGbY+Scqf6ZAcsz+W4vADrULnfDm8RzDfV/NYS+ZfADxa5xAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f35d7b47d9760563cf6aa424980cba4245c71a49e6e426f47077774d6efe7b4","last_reissued_at":"2026-05-18T00:12:35.700805Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:35.700805Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Motzkin numbers and related sequences modulo powers of $2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian Krattenthaler (Universit\\\"at Wien), Thomas W. M\\\"uller (Queen Mary & Westfield College, University of London)","submitted_at":"2016-08-19T16:27:45Z","abstract_excerpt":"We show that the generating function $\\sum_{n\\ge0}M_n\\,z^n$ for Motzkin numbers $M_n$, when coefficients are reduced modulo a given power of $2$, can be expressed as a polynomial in the basic series $\\sum _{e\\ge0} ^{} {z^{4^e}}/( {1-z^{2\\cdot 4^e}})$ with coefficients being Laurent polynomials in $z$ and $1-z$. We use this result to determine $M_n$ modulo $8$ in terms of the binary digits of~$n$, thus improving, respectively complementing earlier results by Eu, Liu and Yeh [Europ. J. Combin. 29 (2008), 1449-1466] and by Rowland and Yassawi [J. Th\\'eorie Nombres Bordeaux 27 (2015), 245-288]. An"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.05657","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1608.05657","created_at":"2026-05-18T00:12:35.700870+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.05657v3","created_at":"2026-05-18T00:12:35.700870+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.05657","created_at":"2026-05-18T00:12:35.700870+00:00"},{"alias_kind":"pith_short_12","alias_value":"N425PND5S5QF","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_16","alias_value":"N425PND5S5QFMPHW","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_8","alias_value":"N425PND5","created_at":"2026-05-18T12:30:32.724797+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/N425PND5S5QFMPHWVJBETAGLUQ","json":"https://pith.science/pith/N425PND5S5QFMPHWVJBETAGLUQ.json","graph_json":"https://pith.science/api/pith-number/N425PND5S5QFMPHWVJBETAGLUQ/graph.json","events_json":"https://pith.science/api/pith-number/N425PND5S5QFMPHWVJBETAGLUQ/events.json","paper":"https://pith.science/paper/N425PND5"},"agent_actions":{"view_html":"https://pith.science/pith/N425PND5S5QFMPHWVJBETAGLUQ","download_json":"https://pith.science/pith/N425PND5S5QFMPHWVJBETAGLUQ.json","view_paper":"https://pith.science/paper/N425PND5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.05657&json=true","fetch_graph":"https://pith.science/api/pith-number/N425PND5S5QFMPHWVJBETAGLUQ/graph.json","fetch_events":"https://pith.science/api/pith-number/N425PND5S5QFMPHWVJBETAGLUQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N425PND5S5QFMPHWVJBETAGLUQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N425PND5S5QFMPHWVJBETAGLUQ/action/storage_attestation","attest_author":"https://pith.science/pith/N425PND5S5QFMPHWVJBETAGLUQ/action/author_attestation","sign_citation":"https://pith.science/pith/N425PND5S5QFMPHWVJBETAGLUQ/action/citation_signature","submit_replication":"https://pith.science/pith/N425PND5S5QFMPHWVJBETAGLUQ/action/replication_record"}},"created_at":"2026-05-18T00:12:35.700870+00:00","updated_at":"2026-05-18T00:12:35.700870+00:00"}