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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","authors_text":"Zhi-Wei Sun","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-01T15:59:59Z","title":"Arithmetic properties of Delannoy numbers and Schr\\\"oder numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.00574","kind":"arxiv","version":5},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:658146786edb20a157f94887aa86b9b83353e124f47490af040e8a144a4abd50","target":"record","created_at":"2026-05-18T00:32:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"43edf08b8f4c3046887cf97523d2957db2ccb6359389b5b7162764218891baad","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-01T15:59:59Z","title_canon_sha256":"1f255afcd67fd5ccaf0452d12967bc6dab1e4dc9170010a0d79a760db797db90"},"schema_version":"1.0","source":{"id":"1602.00574","kind":"arxiv","version":5}},"canonical_sha256":"35f333e3a8b473f20ca940adf3661d7914a78eb27cbfef2b8f6f8990ebe02076","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"35f333e3a8b473f20ca940adf3661d7914a78eb27cbfef2b8f6f8990ebe02076","first_computed_at":"2026-05-18T00:32:31.497879Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:32:31.497879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HpHYE4f8tCvpYjnYnpulWNPrVAPaE6IDlo2Vcs/uSZhiU+OeDuyxqwHoKVw4cwli3Jb1WLSbw3rr878X85FWAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:32:31.498513Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.00574","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:658146786edb20a157f94887aa86b9b83353e124f47490af040e8a144a4abd50","sha256:138e629e61fa211ae83291617700b000942a0c595e9ca74d40d6399ee8d399c0"],"state_sha256":"efa75b8097ca93f91e0cc5ebf8805f22437612a93af3e659c34076a12305a818"}