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In this paper we establish the following surprising arithmetic properties of them with $n$ any positive integer: $$\\frac2n\\sum_{k=1}^n(2k+1)M_k^2\\in\\mathbb Z,$$ $$\\frac{n^2(n^2-1)}6\\,\\bigg|\\,\\sum_{k=0}^{n-1}k(k+1)(8k+9)T_kT_{k+1},$$ and also $$\\sum_{k=0}^{n-1}(k+1)(k+2)(2k+3)M_k^23^{n-1-k}=n(n+1)(n+2)M_nM_{n-1}.$$"},"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":"1801.08905","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-26T17:30:51Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"f17c23bd9b4318b623a040cb69ece7af8617dab013cb47d602314951d93b6da2","abstract_canon_sha256":"aa59612fb310e8d3d45f3a451c990bd53a0fe26bcc849233493977d4eccfdbfa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:52:52.454195Z","signature_b64":"x1WrjUx6hfM6bLynsc1lkNAZ3yklPzJIV6ti5U/723D5oZl5gISC0W9WiqRZPu840VF1V4XgKW5a17E6/JYeDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cbf20099480fc1ce38c2ae98c30d637c4280f8adba40573a457826c035954bc4","last_reissued_at":"2026-07-05T03:52:52.453798Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:52:52.453798Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Motzkin numbers and central trinomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Zhi-Wei Sun","submitted_at":"2018-01-26T17:30:51Z","abstract_excerpt":"The Motzkin numbers $M_n=\\sum_{k=0}^n\\binom n{2k}\\binom{2k}k/(k+1)$ $(n=0,1,2,\\ldots)$ and the central trinomial coefficients $T_n$ ($n=0,1,2,\\ldots)$ given by the constant term of $(1+x+x^{-1})^n$, have many combinatorial interpretations. In this paper we establish the following surprising arithmetic properties of them with $n$ any positive integer: $$\\frac2n\\sum_{k=1}^n(2k+1)M_k^2\\in\\mathbb Z,$$ $$\\frac{n^2(n^2-1)}6\\,\\bigg|\\,\\sum_{k=0}^{n-1}k(k+1)(8k+9)T_kT_{k+1},$$ and also $$\\sum_{k=0}^{n-1}(k+1)(k+2)(2k+3)M_k^23^{n-1-k}=n(n+1)(n+2)M_nM_{n-1}.$$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.08905","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1801.08905/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1801.08905","created_at":"2026-07-05T03:52:52.453855+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.08905v3","created_at":"2026-07-05T03:52:52.453855+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.08905","created_at":"2026-07-05T03:52:52.453855+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZPZABGKIB7A4","created_at":"2026-07-05T03:52:52.453855+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZPZABGKIB7A44OGC","created_at":"2026-07-05T03:52:52.453855+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZPZABGKI","created_at":"2026-07-05T03:52:52.453855+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/ZPZABGKIB7A44OGCV2MMGDLDPR","json":"https://pith.science/pith/ZPZABGKIB7A44OGCV2MMGDLDPR.json","graph_json":"https://pith.science/api/pith-number/ZPZABGKIB7A44OGCV2MMGDLDPR/graph.json","events_json":"https://pith.science/api/pith-number/ZPZABGKIB7A44OGCV2MMGDLDPR/events.json","paper":"https://pith.science/paper/ZPZABGKI"},"agent_actions":{"view_html":"https://pith.science/pith/ZPZABGKIB7A44OGCV2MMGDLDPR","download_json":"https://pith.science/pith/ZPZABGKIB7A44OGCV2MMGDLDPR.json","view_paper":"https://pith.science/paper/ZPZABGKI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.08905&json=true","fetch_graph":"https://pith.science/api/pith-number/ZPZABGKIB7A44OGCV2MMGDLDPR/graph.json","fetch_events":"https://pith.science/api/pith-number/ZPZABGKIB7A44OGCV2MMGDLDPR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZPZABGKIB7A44OGCV2MMGDLDPR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZPZABGKIB7A44OGCV2MMGDLDPR/action/storage_attestation","attest_author":"https://pith.science/pith/ZPZABGKIB7A44OGCV2MMGDLDPR/action/author_attestation","sign_citation":"https://pith.science/pith/ZPZABGKIB7A44OGCV2MMGDLDPR/action/citation_signature","submit_replication":"https://pith.science/pith/ZPZABGKIB7A44OGCV2MMGDLDPR/action/replication_record"}},"created_at":"2026-07-05T03:52:52.453855+00:00","updated_at":"2026-07-05T03:52:52.453855+00:00"}