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We prove that this classical result is the $\\nu=0$ case of an infinite family of ``pentagonal number'' recurrences. For each $\\nu\\geq 0,$ we prove for positive $n$ that\n  $$ p(n)=\\frac{1}{g_{\\nu}(n,0)}\\left(\\alpha_{\\nu}\\cdot \\sigma_{2\\nu-1}(n)+ \\mathrm{Tr}_{2\\nu}(n) +\\sum_{k\\in \\mathbb{Z}\\setminus \\{0\\}} (-1)^{k+1} g_{\\nu}(n,k)\\cd"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2411.16968","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-11-25T22:37:34Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e86ea74c77aaecc65adc63d1d592fa3dee8be1e5e661baae77c2acc10130272d","abstract_canon_sha256":"1c63601f1a47979e5997b20bea576bda9201c1359e0eaf4eaf87c99a304bd854"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:51:28.621255Z","signature_b64":"rJJG27S7NeonBrCI+cUMa0ovcupHAhMHObtsbWsyQmadgK0NKfdsfJUw2048PKv6cjHbutQL+qKwqiQvutQEBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"45f63ab27970d4d3d25f32f1447b2ee576392bb21b9951be6537bfb29c0275c5","last_reissued_at":"2026-07-05T10:51:28.620719Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:51:28.620719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Pentagonal number recurrence relations for $p(n)$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Ajit Singh, Hasan Saad, Ken Ono, Kevin Gomez","submitted_at":"2024-11-25T22:37:34Z","abstract_excerpt":"We revisit Euler's partition function recurrence, which asserts, for integers $n\\geq 1,$ that $$ p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\\dots = \\sum_{k\\in \\mathbb{Z}\\setminus \\{0\\}} (-1)^{k+1} p(n-\\omega(k)), $$ where $\\omega(m):=(3m^2+m)/2$ is the $m$th pentagonal number. We prove that this classical result is the $\\nu=0$ case of an infinite family of ``pentagonal number'' recurrences. For each $\\nu\\geq 0,$ we prove for positive $n$ that\n  $$ p(n)=\\frac{1}{g_{\\nu}(n,0)}\\left(\\alpha_{\\nu}\\cdot \\sigma_{2\\nu-1}(n)+ \\mathrm{Tr}_{2\\nu}(n) +\\sum_{k\\in \\mathbb{Z}\\setminus \\{0\\}} (-1)^{k+1} g_{\\nu}(n,k)\\cd"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.16968","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.16968/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2411.16968","created_at":"2026-07-05T10:51:28.620786+00:00"},{"alias_kind":"arxiv_version","alias_value":"2411.16968v2","created_at":"2026-07-05T10:51:28.620786+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.16968","created_at":"2026-07-05T10:51:28.620786+00:00"},{"alias_kind":"pith_short_12","alias_value":"IX3DVMTZODKN","created_at":"2026-07-05T10:51:28.620786+00:00"},{"alias_kind":"pith_short_16","alias_value":"IX3DVMTZODKNHUS7","created_at":"2026-07-05T10:51:28.620786+00:00"},{"alias_kind":"pith_short_8","alias_value":"IX3DVMTZ","created_at":"2026-07-05T10:51:28.620786+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2412.14344","citing_title":"Euler-type recurrences for $t$-color and $t$-regular partition functions","ref_index":7,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IX3DVMTZODKNHUS7GLYUI6ZO4V","json":"https://pith.science/pith/IX3DVMTZODKNHUS7GLYUI6ZO4V.json","graph_json":"https://pith.science/api/pith-number/IX3DVMTZODKNHUS7GLYUI6ZO4V/graph.json","events_json":"https://pith.science/api/pith-number/IX3DVMTZODKNHUS7GLYUI6ZO4V/events.json","paper":"https://pith.science/paper/IX3DVMTZ"},"agent_actions":{"view_html":"https://pith.science/pith/IX3DVMTZODKNHUS7GLYUI6ZO4V","download_json":"https://pith.science/pith/IX3DVMTZODKNHUS7GLYUI6ZO4V.json","view_paper":"https://pith.science/paper/IX3DVMTZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2411.16968&json=true","fetch_graph":"https://pith.science/api/pith-number/IX3DVMTZODKNHUS7GLYUI6ZO4V/graph.json","fetch_events":"https://pith.science/api/pith-number/IX3DVMTZODKNHUS7GLYUI6ZO4V/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IX3DVMTZODKNHUS7GLYUI6ZO4V/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IX3DVMTZODKNHUS7GLYUI6ZO4V/action/storage_attestation","attest_author":"https://pith.science/pith/IX3DVMTZODKNHUS7GLYUI6ZO4V/action/author_attestation","sign_citation":"https://pith.science/pith/IX3DVMTZODKNHUS7GLYUI6ZO4V/action/citation_signature","submit_replication":"https://pith.science/pith/IX3DVMTZODKNHUS7GLYUI6ZO4V/action/replication_record"}},"created_at":"2026-07-05T10:51:28.620786+00:00","updated_at":"2026-07-05T10:51:28.620786+00:00"}