Volodin, Nikolai Number of multinomial coefficients not divisible by a prime. (English) Zbl 0816.11011 Fibonacci Q. 32, No. 5, 402-406 (1994). For \(n,\ell\in \mathbb{N}\) fixed, a prime \(p\) and \(N\geq 1\) let \(g(n, \ell, p^ N)\) be the number of multinomial coefficients \({{n!} \over {\dot x_ 1! \cdots \dot x_ \ell!}}\) (\(\dot x_ i\geq 0\), \(i=1,\dots, \ell\); \(\dot x_ 1+ \dots+ \dot x_ \ell=n\)) which are not divisible by \(p^ N\) and \(G(n, \ell, p^ N):= \sum_{k=1}^{n-1} g(n, \ell, p^ N)\). Extending investigations of Harborth in the case \(\ell=2\), \(p^ N=2\) it is shown first that \(\limsup_{n\to\infty}\) \(G(n,\ell, p)/ n^ \theta=1\) \((\theta:= \log_ p (\ell, p-1))\), and further that if \(q_ r:= G(n_ r, \ell, 2)/ n_ r^ \theta\) then \((q_ n )_{n\geq 1}\) is strictly decreasing if \(n_ 0=1\) and \(n_{r+1}= 2n_ r \pm 1\) where \(+\) or \(-\) is chosen such that \(q_{r+1}\) becomes minimal. Reviewer: G.Larcher (Salzburg) Cited in 1 Document MSC: 11B65 Binomial coefficients; factorials; \(q\)-identities 11B50 Sequences (mod \(m\)) 05A10 Factorials, binomial coefficients, combinatorial functions Keywords:Pascal’s triangle; number of multinomial coefficients PDFBibTeX XMLCite \textit{N. Volodin}, Fibonacci Q. 32, No. 5, 402--406 (1994; Zbl 0816.11011)