{"paper":{"title":"2-adic Valuations of Coefficients of the Fifth and Ninth Powers of the Thue--Morse Generating Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xinping Wang, Zhao Shen","submitted_at":"2026-06-27T03:52:52Z","abstract_excerpt":"Let $T(x)=\\prod_{k=0}^{\\infty}(1-x^{2^k})$ be the generating function of the Thue--Morse sequence, and write $T(x)^m=\\sum_{n\\geq 0}t_m(n)x^n$. We prove exact formulas for the $2$-adic valuations of the coefficients $t_5(n)$ and $t_9(n)$: \\[ \\nu_2\\bigl(t_5(4n+j)\\bigr)\n  =4\\Bigl\\lceil\\tfrac{\\nu_2(n+1)}{2}\\Bigr\\rceil-\\bigl(\\nu_2(n+1)\\bmod 2\\bigr), \\quad j\\in\\{0,1,2,3\\}, \\] \\[ \\nu_2\\bigl(t_9(8n+j)\\bigr)\n  =5\\Bigl\\lceil\\tfrac{\\nu_2(n+1)}{2}\\Bigr\\rceil-2\\bigl(\\nu_2(n+1)\\bmod 2\\bigr), \\quad j\\in\\{0,1,\\ldots,7\\}. \\] These formulas confirm Conjecture~5.2 of Gawron--Miska--Ulas~\\cite{ga} for $m=5$ and $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28718","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/2606.28718/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"}