{"paper":{"title":"New congruences for sums involving Apery numbers or central Delannoy numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Jiang Zeng, Victor J. W. Guo","submitted_at":"2010-08-17T13:41:05Z","abstract_excerpt":"The Ap\\'ery numbers $A_n$ and central Delannoy numbers $D_n$ are defined by $$A_n=\\sum_{k=0}^{n}{n+k\\choose 2k}^2{2k\\choose k}^2, \\quad D_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k}. $$ Motivated by some recent work of Z.-W. Sun, we prove the following congruences:\n\\sum_{k=0}^{n-1}(2k+1)^{2r+1}A_k &\\equiv \\sum_{k=0}^{n-1}\\varepsilon^k (2k+1)^{2r+1}D_k \\equiv 0\\pmod n,\nwhere $n\\geqslant 1$, $r\\geqslant 0$, and $\\varepsilon=\\pm1$. For $r=1$, we further show that\n\\sum_{k=0}^{n-1}(2k+1)^{3}A_k &\\equiv 0\\pmod{n^3}, \\quad\n\\sum_{k=0}^{p-1}(2k+1)^{3}A_k &\\equiv p^3 \\pmod{2p^6},\nwhere $p>3$ is a prime"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.2894","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"}