{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:GXZTHY5IWRZ7EDFJICW7GZQ5PE","short_pith_number":"pith:GXZTHY5I","schema_version":"1.0","canonical_sha256":"35f333e3a8b473f20ca940adf3661d7914a78eb27cbfef2b8f6f8990ebe02076","source":{"kind":"arxiv","id":"1602.00574","version":5},"attestation_state":"computed","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"},"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":"1602.00574","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-01T15:59:59Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"1f255afcd67fd5ccaf0452d12967bc6dab1e4dc9170010a0d79a760db797db90","abstract_canon_sha256":"43edf08b8f4c3046887cf97523d2957db2ccb6359389b5b7162764218891baad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:31.498513Z","signature_b64":"HpHYE4f8tCvpYjnYnpulWNPrVAPaE6IDlo2Vcs/uSZhiU+OeDuyxqwHoKVw4cwli3Jb1WLSbw3rr878X85FWAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"35f333e3a8b473f20ca940adf3661d7914a78eb27cbfef2b8f6f8990ebe02076","last_reissued_at":"2026-05-18T00:32:31.497879Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:31.497879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1602.00574","created_at":"2026-05-18T00:32:31.497986+00:00"},{"alias_kind":"arxiv_version","alias_value":"1602.00574v5","created_at":"2026-05-18T00:32:31.497986+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.00574","created_at":"2026-05-18T00:32:31.497986+00:00"},{"alias_kind":"pith_short_12","alias_value":"GXZTHY5IWRZ7","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_16","alias_value":"GXZTHY5IWRZ7EDFJ","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_8","alias_value":"GXZTHY5I","created_at":"2026-05-18T12:30:19.053100+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/GXZTHY5IWRZ7EDFJICW7GZQ5PE","json":"https://pith.science/pith/GXZTHY5IWRZ7EDFJICW7GZQ5PE.json","graph_json":"https://pith.science/api/pith-number/GXZTHY5IWRZ7EDFJICW7GZQ5PE/graph.json","events_json":"https://pith.science/api/pith-number/GXZTHY5IWRZ7EDFJICW7GZQ5PE/events.json","paper":"https://pith.science/paper/GXZTHY5I"},"agent_actions":{"view_html":"https://pith.science/pith/GXZTHY5IWRZ7EDFJICW7GZQ5PE","download_json":"https://pith.science/pith/GXZTHY5IWRZ7EDFJICW7GZQ5PE.json","view_paper":"https://pith.science/paper/GXZTHY5I","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1602.00574&json=true","fetch_graph":"https://pith.science/api/pith-number/GXZTHY5IWRZ7EDFJICW7GZQ5PE/graph.json","fetch_events":"https://pith.science/api/pith-number/GXZTHY5IWRZ7EDFJICW7GZQ5PE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GXZTHY5IWRZ7EDFJICW7GZQ5PE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GXZTHY5IWRZ7EDFJICW7GZQ5PE/action/storage_attestation","attest_author":"https://pith.science/pith/GXZTHY5IWRZ7EDFJICW7GZQ5PE/action/author_attestation","sign_citation":"https://pith.science/pith/GXZTHY5IWRZ7EDFJICW7GZQ5PE/action/citation_signature","submit_replication":"https://pith.science/pith/GXZTHY5IWRZ7EDFJICW7GZQ5PE/action/replication_record"}},"created_at":"2026-05-18T00:32:31.497986+00:00","updated_at":"2026-05-18T00:32:31.497986+00:00"}