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By interpreting both families as special values of weighted Stirling polynomials, we answer Lehmer's question of whether $B(n,k)$ satisfies a recurrence with a fixed number of terms by proving the four-term recurrence $kB(n+1,k+1)=B(n,k-1)-(n-k)B(n,k)-B(n+1,k)$. The corresponding relation for the unsigned array was conjectured by M. Kurkov, but not proved, in OEIS A354794. We further derive explicit formulas, convolution identities, and"},"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":"2607.26613","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-29T08:38:59Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b662e39c894f2cb4cb82a8e26cf256f83694b74f10ddf88253c23f19b85544cf","abstract_canon_sha256":"2bd26a998a0788a10b3c46df4f05dd0d1ee25cb8d7826023112ccda3c5413dcf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5979755d9eb87eb115ca19bcf01d4ec6c02be4f20e304eedd8f6ba1808071189","last_reissued_at":"2026-07-30T01:21:34.486764Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-30T01:21:34.486764Z"},"graph_snapshot":{"paper":{"title":"On a Question of Lehmer concerning the Comtet Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Sangtae Jeong","submitted_at":"2026-07-29T08:38:59Z","abstract_excerpt":"The Comtet numbers $b(n,k)$ and $B(n,k)$ of the first and second kind arise from the powers of $(1+x)\\log(1+x)$ and of its compositional inverse, respectively. By interpreting both families as special values of weighted Stirling polynomials, we answer Lehmer's question of whether $B(n,k)$ satisfies a recurrence with a fixed number of terms by proving the four-term recurrence $kB(n+1,k+1)=B(n,k-1)-(n-k)B(n,k)-B(n+1,k)$. The corresponding relation for the unsigned array was conjectured by M. Kurkov, but not proved, in OEIS A354794. 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