{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2005:DMVKW6GXTIPVVNJ5XHSSMHZF73","short_pith_number":"pith:DMVKW6GX","schema_version":"1.0","canonical_sha256":"1b2aab78d79a1f5ab53db9e5261f25fec02134f5ea58a0836b9bc399d5ff9972","source":{"kind":"arxiv","id":"math/0509648","version":9},"attestation_state":"computed","paper":{"title":"A combinatorial identity with application to Catalan numbers","license":"","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Hao Pan, Zhi-Wei Sun","submitted_at":"2005-09-27T22:24:05Z","abstract_excerpt":"By a very simple argument, we prove that if $l,m,n$ are nonnegative integers then $$\\sum_{k=0}^l(-1)^{m-k}\\binom{l}{k}\\binom{m-k}{n}\\binom{2k}{k-2l+m} =\\sum_{k=0}^l\\binom{l}{k}\\binom{2k}{n}\\binom{n-l}{m+n-3k-l}.\n  On the basis of this identity, for $d,r=0,1,2,...$ we construct explicit $F(d,r)$ and $G(d,r)$ such that for any prime $p>\\max\\{d,r\\}$ we have\n  \\sum_{k=1}^{p-1}k^r C_{k+d}\\equiv \\cases F(d,r)(mod p)& if 3|p-1, \\\\G(d,r)\\ (mod p)& if 3|p-2,\n  where $C_n$ denotes the Catalan number $(n+1)^{-1}\\binom{2n}{n}$. For example, when $p\\geq 5$ is a prime, we have\n  \\sum_{k=1}^{p-1}k^2C_k\\equiv"},"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":"math/0509648","kind":"arxiv","version":9},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2005-09-27T22:24:05Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"07fa58d1c9bcfc30b5d6d09145a5e49ca69076e2343f0f6822b8c693369edaf5","abstract_canon_sha256":"d35c8ef2bd692cccf009c71a4d6fe1c25eb6b7f6db8b8b60c5ecb794c84638a9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:50:38.788127Z","signature_b64":"VNYRkrp3nJxTQiDO5m0OcUkhDFN9/9mndBt/tskDCi+Tm1OenOMwVnZxpCDbT9Vajdr612kscSf0qQyKPDhFAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b2aab78d79a1f5ab53db9e5261f25fec02134f5ea58a0836b9bc399d5ff9972","last_reissued_at":"2026-07-04T14:50:38.787763Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:50:38.787763Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A combinatorial identity with application to Catalan numbers","license":"","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Hao Pan, Zhi-Wei Sun","submitted_at":"2005-09-27T22:24:05Z","abstract_excerpt":"By a very simple argument, we prove that if $l,m,n$ are nonnegative integers then $$\\sum_{k=0}^l(-1)^{m-k}\\binom{l}{k}\\binom{m-k}{n}\\binom{2k}{k-2l+m} =\\sum_{k=0}^l\\binom{l}{k}\\binom{2k}{n}\\binom{n-l}{m+n-3k-l}.\n  On the basis of this identity, for $d,r=0,1,2,...$ we construct explicit $F(d,r)$ and $G(d,r)$ such that for any prime $p>\\max\\{d,r\\}$ we have\n  \\sum_{k=1}^{p-1}k^r C_{k+d}\\equiv \\cases F(d,r)(mod p)& if 3|p-1, \\\\G(d,r)\\ (mod p)& if 3|p-2,\n  where $C_n$ denotes the Catalan number $(n+1)^{-1}\\binom{2n}{n}$. For example, when $p\\geq 5$ is a prime, we have\n  \\sum_{k=1}^{p-1}k^2C_k\\equiv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0509648","kind":"arxiv","version":9},"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/math/0509648/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":"math/0509648","created_at":"2026-07-04T14:50:38.787826+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0509648v9","created_at":"2026-07-04T14:50:38.787826+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0509648","created_at":"2026-07-04T14:50:38.787826+00:00"},{"alias_kind":"pith_short_12","alias_value":"DMVKW6GXTIPV","created_at":"2026-07-04T14:50:38.787826+00:00"},{"alias_kind":"pith_short_16","alias_value":"DMVKW6GXTIPVVNJ5","created_at":"2026-07-04T14:50:38.787826+00:00"},{"alias_kind":"pith_short_8","alias_value":"DMVKW6GX","created_at":"2026-07-04T14:50:38.787826+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/DMVKW6GXTIPVVNJ5XHSSMHZF73","json":"https://pith.science/pith/DMVKW6GXTIPVVNJ5XHSSMHZF73.json","graph_json":"https://pith.science/api/pith-number/DMVKW6GXTIPVVNJ5XHSSMHZF73/graph.json","events_json":"https://pith.science/api/pith-number/DMVKW6GXTIPVVNJ5XHSSMHZF73/events.json","paper":"https://pith.science/paper/DMVKW6GX"},"agent_actions":{"view_html":"https://pith.science/pith/DMVKW6GXTIPVVNJ5XHSSMHZF73","download_json":"https://pith.science/pith/DMVKW6GXTIPVVNJ5XHSSMHZF73.json","view_paper":"https://pith.science/paper/DMVKW6GX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0509648&json=true","fetch_graph":"https://pith.science/api/pith-number/DMVKW6GXTIPVVNJ5XHSSMHZF73/graph.json","fetch_events":"https://pith.science/api/pith-number/DMVKW6GXTIPVVNJ5XHSSMHZF73/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DMVKW6GXTIPVVNJ5XHSSMHZF73/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DMVKW6GXTIPVVNJ5XHSSMHZF73/action/storage_attestation","attest_author":"https://pith.science/pith/DMVKW6GXTIPVVNJ5XHSSMHZF73/action/author_attestation","sign_citation":"https://pith.science/pith/DMVKW6GXTIPVVNJ5XHSSMHZF73/action/citation_signature","submit_replication":"https://pith.science/pith/DMVKW6GXTIPVVNJ5XHSSMHZF73/action/replication_record"}},"created_at":"2026-07-04T14:50:38.787826+00:00","updated_at":"2026-07-04T14:50:38.787826+00:00"}