{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:WIKANR3DVQ4OO6DBDENBT2G3OQ","short_pith_number":"pith:WIKANR3D","schema_version":"1.0","canonical_sha256":"b21406c763ac38e77861191a19e8db741122569e8da01ff2f247aa65bc87f492","source":{"kind":"arxiv","id":"2605.16553","version":1},"attestation_state":"computed","paper":{"title":"An explicit algebraic generating function for OEIS A348410","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tong Niu","submitted_at":"2026-05-15T18:53:57Z","abstract_excerpt":"For the OEIS sequence A348410, P. Bala recorded in February 2022 two equivalent closed forms, $a(n) = [x^{n}] ((1-x)(1-x^2))^{-n}$ and a single-index binomial sum. R. J. Mathar (October 2021) and V. Kotesovec (November 2021) each contributed a conjectured P-recursive recurrence -- Mathar's of order $4$, Kotesovec's of order $2$. We apply Lagrange-B\\\"urmann inversion to Bala's $[x^n]$ form to derive the parametric expression $A(t) = (1 - y^2)/(1 - y - 4 y^2)$, where $y = y(t)$ is implicit by $y(1-y)^2(1+y) = t$. Eliminating $y$ via resultant gives the explicit algebraic equation $P(t, A) = 0$ o"},"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":"2605.16553","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-15T18:53:57Z","cross_cats_sorted":[],"title_canon_sha256":"58579f2132c99b7d24539ec0d4497883b7be525ceeebd33d49576cdeaa4276c1","abstract_canon_sha256":"e069f37e99f781a1e0adc513f7f38d8c98f0c9cfea43344ccc7fc60667fc4de1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:02:28.851331Z","signature_b64":"kwSPsHC2F/3mSBkDOjJizUhu8gBJ8k20wpH3pc892UMDz/MB2rW0Q7oHrW5mkC3+DANArfYtunxQlCwLUNVoCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b21406c763ac38e77861191a19e8db741122569e8da01ff2f247aa65bc87f492","last_reissued_at":"2026-05-20T00:02:28.850487Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:02:28.850487Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An explicit algebraic generating function for OEIS A348410","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tong Niu","submitted_at":"2026-05-15T18:53:57Z","abstract_excerpt":"For the OEIS sequence A348410, P. Bala recorded in February 2022 two equivalent closed forms, $a(n) = [x^{n}] ((1-x)(1-x^2))^{-n}$ and a single-index binomial sum. R. J. Mathar (October 2021) and V. Kotesovec (November 2021) each contributed a conjectured P-recursive recurrence -- Mathar's of order $4$, Kotesovec's of order $2$. We apply Lagrange-B\\\"urmann inversion to Bala's $[x^n]$ form to derive the parametric expression $A(t) = (1 - y^2)/(1 - y - 4 y^2)$, where $y = y(t)$ is implicit by $y(1-y)^2(1+y) = t$. Eliminating $y$ via resultant gives the explicit algebraic equation $P(t, A) = 0$ o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.16553","kind":"arxiv","version":1},"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/2605.16553/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-19T19:21:56.895641Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.632128Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"8873125bef705c0e5fe83c02a1b5fe27127418f4c6251ac0abfc879e5156e3eb"},"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":"2605.16553","created_at":"2026-05-20T00:02:28.850621+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.16553v1","created_at":"2026-05-20T00:02:28.850621+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.16553","created_at":"2026-05-20T00:02:28.850621+00:00"},{"alias_kind":"pith_short_12","alias_value":"WIKANR3DVQ4O","created_at":"2026-05-20T00:02:28.850621+00:00"},{"alias_kind":"pith_short_16","alias_value":"WIKANR3DVQ4OO6DB","created_at":"2026-05-20T00:02:28.850621+00:00"},{"alias_kind":"pith_short_8","alias_value":"WIKANR3D","created_at":"2026-05-20T00:02:28.850621+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2605.21255","citing_title":"The generating function of A348410 in OEIS using the diagonal method and another sequence (A001008) from OEIS","ref_index":4,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WIKANR3DVQ4OO6DBDENBT2G3OQ","json":"https://pith.science/pith/WIKANR3DVQ4OO6DBDENBT2G3OQ.json","graph_json":"https://pith.science/api/pith-number/WIKANR3DVQ4OO6DBDENBT2G3OQ/graph.json","events_json":"https://pith.science/api/pith-number/WIKANR3DVQ4OO6DBDENBT2G3OQ/events.json","paper":"https://pith.science/paper/WIKANR3D"},"agent_actions":{"view_html":"https://pith.science/pith/WIKANR3DVQ4OO6DBDENBT2G3OQ","download_json":"https://pith.science/pith/WIKANR3DVQ4OO6DBDENBT2G3OQ.json","view_paper":"https://pith.science/paper/WIKANR3D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.16553&json=true","fetch_graph":"https://pith.science/api/pith-number/WIKANR3DVQ4OO6DBDENBT2G3OQ/graph.json","fetch_events":"https://pith.science/api/pith-number/WIKANR3DVQ4OO6DBDENBT2G3OQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WIKANR3DVQ4OO6DBDENBT2G3OQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WIKANR3DVQ4OO6DBDENBT2G3OQ/action/storage_attestation","attest_author":"https://pith.science/pith/WIKANR3DVQ4OO6DBDENBT2G3OQ/action/author_attestation","sign_citation":"https://pith.science/pith/WIKANR3DVQ4OO6DBDENBT2G3OQ/action/citation_signature","submit_replication":"https://pith.science/pith/WIKANR3DVQ4OO6DBDENBT2G3OQ/action/replication_record"}},"created_at":"2026-05-20T00:02:28.850621+00:00","updated_at":"2026-05-20T00:02:28.850621+00:00"}