{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:W7K4DU5XUQCGEGD5LX65D4RCCY","short_pith_number":"pith:W7K4DU5X","schema_version":"1.0","canonical_sha256":"b7d5c1d3b7a40462187d5dfdd1f222162a2d86b40f727b5bc7a94241105f652c","source":{"kind":"arxiv","id":"1009.2486","version":4},"attestation_state":"computed","paper":{"title":"On Delannoy numbers and Schr\\\"oder numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2010-09-13T19:36:16Z","abstract_excerpt":"The n-th Delannoy number and the n-th Schr\\\"oder number given by $D_n=\\sum_{k=0}^n\\binom{n}{k}\\binom{n+k}{k}$ and $S_n=\\sum_{k=0}^n\\binom{n}{k}\\binom{n+k}{k}/(k+1)$ respectively arise naturally from enumerative combinatorics. 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"},"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":"1009.2486","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-09-13T19:36:16Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"3643d143ca9b67bdbb43869b34e97e9ffb11120d20c5a2ba8158785d694fa89a","abstract_canon_sha256":"ec0abf01884c77c4297b150cb5bbd30580a7568eb0761984fe70da8dfe272539"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:04.373776Z","signature_b64":"1ZlZY8nCzJrHYaBj3xGw7JZqdZuq1RJLEVZY/U+ypFfS+KVcY02op64gBX0cltcqiRLda4FGmJrBaHyjgt/sDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b7d5c1d3b7a40462187d5dfdd1f222162a2d86b40f727b5bc7a94241105f652c","last_reissued_at":"2026-05-18T04:15:04.373061Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:04.373061Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Delannoy numbers and Schr\\\"oder numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2010-09-13T19:36:16Z","abstract_excerpt":"The n-th Delannoy number and the n-th Schr\\\"oder number given by $D_n=\\sum_{k=0}^n\\binom{n}{k}\\binom{n+k}{k}$ and $S_n=\\sum_{k=0}^n\\binom{n}{k}\\binom{n+k}{k}/(k+1)$ respectively arise naturally from enumerative combinatorics. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2486","kind":"arxiv","version":4},"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":"1009.2486","created_at":"2026-05-18T04:15:04.373186+00:00"},{"alias_kind":"arxiv_version","alias_value":"1009.2486v4","created_at":"2026-05-18T04:15:04.373186+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.2486","created_at":"2026-05-18T04:15:04.373186+00:00"},{"alias_kind":"pith_short_12","alias_value":"W7K4DU5XUQCG","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_16","alias_value":"W7K4DU5XUQCGEGD5","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_8","alias_value":"W7K4DU5X","created_at":"2026-05-18T12:26:15.391820+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/W7K4DU5XUQCGEGD5LX65D4RCCY","json":"https://pith.science/pith/W7K4DU5XUQCGEGD5LX65D4RCCY.json","graph_json":"https://pith.science/api/pith-number/W7K4DU5XUQCGEGD5LX65D4RCCY/graph.json","events_json":"https://pith.science/api/pith-number/W7K4DU5XUQCGEGD5LX65D4RCCY/events.json","paper":"https://pith.science/paper/W7K4DU5X"},"agent_actions":{"view_html":"https://pith.science/pith/W7K4DU5XUQCGEGD5LX65D4RCCY","download_json":"https://pith.science/pith/W7K4DU5XUQCGEGD5LX65D4RCCY.json","view_paper":"https://pith.science/paper/W7K4DU5X","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1009.2486&json=true","fetch_graph":"https://pith.science/api/pith-number/W7K4DU5XUQCGEGD5LX65D4RCCY/graph.json","fetch_events":"https://pith.science/api/pith-number/W7K4DU5XUQCGEGD5LX65D4RCCY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/W7K4DU5XUQCGEGD5LX65D4RCCY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/W7K4DU5XUQCGEGD5LX65D4RCCY/action/storage_attestation","attest_author":"https://pith.science/pith/W7K4DU5XUQCGEGD5LX65D4RCCY/action/author_attestation","sign_citation":"https://pith.science/pith/W7K4DU5XUQCGEGD5LX65D4RCCY/action/citation_signature","submit_replication":"https://pith.science/pith/W7K4DU5XUQCGEGD5LX65D4RCCY/action/replication_record"}},"created_at":"2026-05-18T04:15:04.373186+00:00","updated_at":"2026-05-18T04:15:04.373186+00:00"}