{"paper":{"title":"Arithmetic properties of coefficients of power series expansion of $\\prod_{n=0}^{\\infty}\\left(1-x^{2^{n}}\\right)^{t}$ (with an Appendix by Andrzej Schinzel)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Maciej Gawron, Maciej Ulas, Piotr Miska","submitted_at":"2017-03-06T16:28:31Z","abstract_excerpt":"Let $F(x)=\\prod_{n=0}^{\\infty}(1-x^{2^{n}})$ be the generating function for the Prouhet-Thue-Morse sequence $((-1)^{s_{2}(n)})_{n\\in\\N}$. In this paper we initiate the study of the arithmetic properties of coefficients of the power series expansions of the function $$ F_{t}(x)=F(x)^{t}=\\sum_{n=0}^{\\infty}f_{n}(t)x^{n}. $$ For $t\\in\\N_{+}$ the sequence $(f_{n}(t))_{n\\in\\N}$ is the Cauchy convolution of $t$ copies of the Prouhet-Thue-Morse sequence. For $t\\in\\Z_{<0}$ the $n$-th term of the sequence $(f_{n}(t))_{n\\in\\N}$ counts the number of representations of the number $n$ as a sum of powers of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.01955","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":""},"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"}