{"paper":{"title":"An explicit generating function arising in counting binomial coefficients divisible by powers of primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Lukas Spiegelhofer, Michael Wallner","submitted_at":"2016-04-24T22:33:46Z","abstract_excerpt":"For a prime $p$ and nonnegative integers $j$ and $n$ let $\\vartheta_p(j,n)$ be the number of entries in the $n$-th row of Pascal's triangle that are exactly divisible by $p^j$. Moreover, for a finite sequence $w=(w_{r-1}\\cdots w_0)\\neq (0,\\ldots,0)$ in $\\{0,\\ldots,p-1\\}$ we denote by $\\lvert n\\rvert_w$ the number of times that $w$ appears as a factor (contiguous subsequence) of the base-$p$ expansion $n=(n_{\\mu-1}\\cdots n_0)_p$ of $n$. It follows from the work of Barat and Grabner (Digital functions and distribution of binomial coefficients, J. London Math. Soc. (2) 64(3), 2001), that $\\varthe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.07089","kind":"arxiv","version":5},"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"}