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Let p be an odd prime. We mainly show that $$\\sum_{k=1}^{p-1}D_k/k^2=2(-1/p)E_{p-3} (mod p)$$ and $$\\sum_{k=1}^{p-1}S_k/m^k=(m^2-6m+1)/(2m)*(1-((m^2-6m+1)/p) (mod p),$$ where (-) is the Legendre symbol, E_0,E_1,E_2,... are Euler numbers and m is any integer not divisible by p. We also conjecture that $\\sum_{k=1}^{p-1}D_k^2/k^2=-2q_p(2)^2 (mod p)$, where $q_p(2)=(2^{p-1}-1","authors_text":"Zhi-Wei Sun","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-09-13T19:36:16Z","title":"On Delannoy numbers and Schr\\\"oder numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2486","kind":"arxiv","version":4},"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:d3b3d948e09c7f9d7316866e76ad23781b19e2c54df0cc1d442e46ea65bd2cf9","target":"record","created_at":"2026-05-18T04:15:04Z","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":"ec0abf01884c77c4297b150cb5bbd30580a7568eb0761984fe70da8dfe272539","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-09-13T19:36:16Z","title_canon_sha256":"3643d143ca9b67bdbb43869b34e97e9ffb11120d20c5a2ba8158785d694fa89a"},"schema_version":"1.0","source":{"id":"1009.2486","kind":"arxiv","version":4}},"canonical_sha256":"b7d5c1d3b7a40462187d5dfdd1f222162a2d86b40f727b5bc7a94241105f652c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b7d5c1d3b7a40462187d5dfdd1f222162a2d86b40f727b5bc7a94241105f652c","first_computed_at":"2026-05-18T04:15:04.373061Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:15:04.373061Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1ZlZY8nCzJrHYaBj3xGw7JZqdZuq1RJLEVZY/U+ypFfS+KVcY02op64gBX0cltcqiRLda4FGmJrBaHyjgt/sDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:15:04.373776Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.2486","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d3b3d948e09c7f9d7316866e76ad23781b19e2c54df0cc1d442e46ea65bd2cf9","sha256:de364f6ab0cd392b8a8db3af0027a4713094b99be836b3b35a1afb9f79318f85"],"state_sha256":"1967ec6e9cd6f5cc2f008cd575c396f3aa0f6d9e6ddc339b3eabc0c500163efc"}