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Based on experimental results, it was conjectured that for any $n$, $\\binom {n,q}{cn}-\\binom {n,q-1}{cn}$ is unimodal and its maximum value occurs $q=\\lfloor\\log_{1+\\frac 1{c}}{n}\\rfloor$ or $q=\\lfloor\\log_{1+\\frac 1{c}}{n}\\rfloor+1$. In particular, when $c=1$, its maximum value occurs for $q=\\lfloor\\log_2{n}\\rfloor$ or $q=\\lfloor\\log_"},"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":"1405.1803","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-08T05:18:13Z","cross_cats_sorted":[],"title_canon_sha256":"b018763bc3f9c386b672eb2c87dde893c44b68cc24ab1b0f14e3a509a83e8f2e","abstract_canon_sha256":"ec3de3302606e88a8f8cd84af50371b80f645d0de76b9f4b9d0ed13c67c31837"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:14.002963Z","signature_b64":"83ieS9YjkJlQ5Mvf+C2XUNLBadoGSRF6G9QTwCfP1hpnMHd8jpu5IcaqG2DDcdGNtFItb2KZ8zoVnMmeIqYqBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f7aebfdd0d9f4ba3f61aa850b4514eee9b490011308cbdde117338baa3022432","last_reissued_at":"2026-05-18T02:32:14.002331Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:14.002331Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic estimate for the polynomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jiyou Li","submitted_at":"2014-05-08T05:18:13Z","abstract_excerpt":"The polynomial coefficient $\\binom {n,q}{k}$ is defined to be the coefficient of $x^{k}$ in the expansion of $(1+x+x^2+... +x^{q-1})^n$. In this note we give an asymptotic estimate for $\\binom {n,q}{cn}$ as $n$ tends to infinity, where $c$ is a positive integer. Based on experimental results, it was conjectured that for any $n$, $\\binom {n,q}{cn}-\\binom {n,q-1}{cn}$ is unimodal and its maximum value occurs $q=\\lfloor\\log_{1+\\frac 1{c}}{n}\\rfloor$ or $q=\\lfloor\\log_{1+\\frac 1{c}}{n}\\rfloor+1$. 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