{"paper":{"title":"Motzkin numbers and related sequences modulo powers of $2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian Krattenthaler (Universit\\\"at Wien), Thomas W. M\\\"uller (Queen Mary & Westfield College, University of London)","submitted_at":"2016-08-19T16:27:45Z","abstract_excerpt":"We show that the generating function $\\sum_{n\\ge0}M_n\\,z^n$ for Motzkin numbers $M_n$, when coefficients are reduced modulo a given power of $2$, can be expressed as a polynomial in the basic series $\\sum _{e\\ge0} ^{} {z^{4^e}}/( {1-z^{2\\cdot 4^e}})$ with coefficients being Laurent polynomials in $z$ and $1-z$. We use this result to determine $M_n$ modulo $8$ in terms of the binary digits of~$n$, thus improving, respectively complementing earlier results by Eu, Liu and Yeh [Europ. J. Combin. 29 (2008), 1449-1466] and by Rowland and Yassawi [J. Th\\'eorie Nombres Bordeaux 27 (2015), 245-288]. An"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.05657","kind":"arxiv","version":3},"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"}