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Those numbers $g_n=g_n(1)$ are closely related to Ap\\'ery numbers and Franel numbers. In this paper we establish some fundamental congruences involving $g_n(x)$. For example, for any prime $p>5$ we have $$\\sum_{k=1}^{p-1}\\frac{g_k(-1)}{k}\\equiv 0\\pmod{p^2}\\quad{and}\\quad\\sum_{k=1}^{p-1}\\frac{g_k(-1)}{k^2}\\equiv 0\\pmod p.$$ This is similar to Wolstenholme's classical congruences $$\\sum_{k=1}^{p-1}\\frac1k\\equiv0\\pmod{p^2}\\quad{and}\\quad\\sum_{k=1}^{p-1}\\frac{1}{k^2}\\equiv0\\pmod p$$ for any prime $p>3$.","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":"2014-07-03T15:58:17Z","title":"Congruences involving $g_n(x)=\\sum_{k=0}^n\\binom nk^2\\binom{2k}kx^k$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.0967","kind":"arxiv","version":8},"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":"sha256:3a3b974f35da4de1dcf19adf9f878fe78a1692ab693aec05b27ea7cd4acc2c43","target":"record","created_at":"2026-05-18T01:10:54Z","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":"bbf530715e4676b3cec852634bf7802792c919e5c3f4a147156e80893183f71d","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-03T15:58:17Z","title_canon_sha256":"28e074b30b42740b06160e243e18ffb3e5824a1d88ab6c94e0c8f40f1a1f69dc"},"schema_version":"1.0","source":{"id":"1407.0967","kind":"arxiv","version":8}},"canonical_sha256":"e55bf8b3c6a35a08407ee9c5fbdc4950c9a71ee5362583a213e1ef31eb201f6d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e55bf8b3c6a35a08407ee9c5fbdc4950c9a71ee5362583a213e1ef31eb201f6d","first_computed_at":"2026-05-18T01:10:54.668604Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:10:54.668604Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u35VH/IV7t1RtcetSp3j0K//e3vr/V4K6dRlvBWfYcGGZ+tuVGVz4hlpH8lLO8eNwdTSMIFAf57La4Hm+RoxCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:10:54.669201Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.0967","source_kind":"arxiv","source_version":8}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3a3b974f35da4de1dcf19adf9f878fe78a1692ab693aec05b27ea7cd4acc2c43","sha256:57298f6d34978bdb60d93a4a8d8b51df21ad0ae3b24082fadc4c8fd25aa1aba0"],"state_sha256":"2b18ed0c48719e9dceb237b766c9abee14598103a2fc81991cde12e2a66342be"}