{"paper":{"title":"Congruences for Catalan-Larcombe-French numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Xiao-Juan Ji, Zhi-Hong Sun","submitted_at":"2015-05-04T15:02:27Z","abstract_excerpt":"Let $\\{P_n\\}$ be the Catalan-Larcombe-French numbers given by $P_0=1,\\ P_1=8$ and $n^2P_n=8(3n^2-3n+1)P_{n-1}-128(n-1)^2P_{n-2}$ $(n\\ge 2)$, and let $S_n=P_n/2^n$. In this paper we deduce congruences for $S_{mp^r}\\pmod{p^{r+2}}$, $S_{mp^r-1}\\pmod{p^r}$ and $S_{mp^r+1}\\pmod{p^{2r}}$, where $p$ is an odd prime and $m,r$ are positive integers. We also prove that $S_{(p^2-1)/2}\\equiv 0\\pmod {p^2}$ for any prime $p\\equiv 5,7\\pmod 8$, and show that $\\{S_m\\}$ is log-convex."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.00668","kind":"arxiv","version":2},"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"}