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Guo","submitted_at":"2017-01-24T03:36:08Z","abstract_excerpt":"We show that, for all positive integers $n_1, \\ldots, n_m$, $n_{m+1}=n_1$, and any non-negative integers $j$ and $r$ with $j\\leqslant m$, the expression $$ \\frac{1}{[n_1]}{n_1+n_{m}\\brack n_1}^{-1} \\sum_{k=1}^{n_1}[2k][k]^{2r}q^{jk^2-(r+1)k}\\prod_{i=1}^{m} {n_i+n_{i+1}\\brack n_i+k} $$ is a Laurent polynomial in $q$ with integer cofficients, where $[n]=1+q+\\cdots+q^{n-1}$ and ${n\\brack k}=\\prod_{i=1}^k(1-q^{n-i+1})/(1-q^i)$. This gives a $q$-analogue of a divisibility result on the Catalan triangle obtained by the first author and Zeng, and also confirms a conjecture of the first author and Zen"},"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":"1701.07016","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-24T03:36:08Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"3da8ddc656c386438c68ffe03ce6ea5a1257fe12459903e35ef1dca6af53cacd","abstract_canon_sha256":"5108b2fb991033c5e26d96e989e1737351abeba40b6c96706aa60e4127c5a47e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:12.766035Z","signature_b64":"ZdMDJmzVbwFNff33voGMBNuZBOkNPysrFkMozIKMqaBNsiDSFYhNqWqH+WDIBy3fomqda4ByG7vOtEjlHCGGDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc9b9eabe0b1223026d340c120f699f6dda8162e92aa4dcf73adf8cab72a06bc","last_reissued_at":"2026-05-18T00:39:12.765447Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:12.765447Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Factors of sums involving $q$-binomial coefficients and powers of $q$-integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Su-Dan Wang, Victor J. W. Guo","submitted_at":"2017-01-24T03:36:08Z","abstract_excerpt":"We show that, for all positive integers $n_1, \\ldots, n_m$, $n_{m+1}=n_1$, and any non-negative integers $j$ and $r$ with $j\\leqslant m$, the expression $$ \\frac{1}{[n_1]}{n_1+n_{m}\\brack n_1}^{-1} \\sum_{k=1}^{n_1}[2k][k]^{2r}q^{jk^2-(r+1)k}\\prod_{i=1}^{m} {n_i+n_{i+1}\\brack n_i+k} $$ is a Laurent polynomial in $q$ with integer cofficients, where $[n]=1+q+\\cdots+q^{n-1}$ and ${n\\brack k}=\\prod_{i=1}^k(1-q^{n-i+1})/(1-q^i)$. This gives a $q$-analogue of a divisibility result on the Catalan triangle obtained by the first author and Zeng, and also confirms a conjecture of the first author and Zen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.07016","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":""},"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":"1701.07016","created_at":"2026-05-18T00:39:12.765548+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.07016v2","created_at":"2026-05-18T00:39:12.765548+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.07016","created_at":"2026-05-18T00:39:12.765548+00:00"},{"alias_kind":"pith_short_12","alias_value":"3SNZ5K7AWERD","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_16","alias_value":"3SNZ5K7AWERDAJWT","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_8","alias_value":"3SNZ5K7A","created_at":"2026-05-18T12:30:58.224056+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3SNZ5K7AWERDAJWTIDASB5UZ63","json":"https://pith.science/pith/3SNZ5K7AWERDAJWTIDASB5UZ63.json","graph_json":"https://pith.science/api/pith-number/3SNZ5K7AWERDAJWTIDASB5UZ63/graph.json","events_json":"https://pith.science/api/pith-number/3SNZ5K7AWERDAJWTIDASB5UZ63/events.json","paper":"https://pith.science/paper/3SNZ5K7A"},"agent_actions":{"view_html":"https://pith.science/pith/3SNZ5K7AWERDAJWTIDASB5UZ63","download_json":"https://pith.science/pith/3SNZ5K7AWERDAJWTIDASB5UZ63.json","view_paper":"https://pith.science/paper/3SNZ5K7A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.07016&json=true","fetch_graph":"https://pith.science/api/pith-number/3SNZ5K7AWERDAJWTIDASB5UZ63/graph.json","fetch_events":"https://pith.science/api/pith-number/3SNZ5K7AWERDAJWTIDASB5UZ63/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3SNZ5K7AWERDAJWTIDASB5UZ63/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3SNZ5K7AWERDAJWTIDASB5UZ63/action/storage_attestation","attest_author":"https://pith.science/pith/3SNZ5K7AWERDAJWTIDASB5UZ63/action/author_attestation","sign_citation":"https://pith.science/pith/3SNZ5K7AWERDAJWTIDASB5UZ63/action/citation_signature","submit_replication":"https://pith.science/pith/3SNZ5K7AWERDAJWTIDASB5UZ63/action/replication_record"}},"created_at":"2026-05-18T00:39:12.765548+00:00","updated_at":"2026-05-18T00:39:12.765548+00:00"}