{"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"}