{"paper":{"title":"Arithmetic properties of Delannoy numbers and Schr\\\"oder numbers","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":"2016-02-01T15:59:59Z","abstract_excerpt":"Define $$D_n(x)=\\sum_{k=0}^n\\binom nk^2x^k(x+1)^{n-k}\\ \\ \\ \\mbox{for}\\ n=0,1,2,\\ldots$$ and $$s_n(x)=\\sum_{k=1}^n\\frac1n\\binom nk\\binom n{k-1}x^{k-1}(x+1)^{n-k}\\ \\ \\ \\mbox{for}\\ n=1,2,3,\\ldots.$$ Then $D_n(1)$ is the $n$-th central Delannoy number $D_n$, and $s_n(1)$ is the $n$-th little Schr\\\"oder number $s_n$. In this paper we obtain some surprising arithmetic properties of $D_n(x)$ and $s_n(x)$. We show that $$\\frac1n\\sum_{k=0}^{n-1}D_k(x)s_{k+1}(x)\\in\\mathbb Z[x(x+1)]\\ \\quad\\mbox{for all}\\ n=1,2,3,\\ldots.$$ Moreover, for any odd prime $p$ and $p$-adic integer $x\\not\\equiv0,-1\\pmod p$, we e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.00574","kind":"arxiv","version":5},"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"}