Sylvester's Theorem and the Non-integrality of a certain Binomial Sum

Daniel López-Aguayo, Florian Luca

In this note, we show that $S(n,r):={\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)\frac{k}{k+r}$ is not an integer for any positive integer $n$ and $r\in \{1,2,3,4,5,6\}$ and for $n\le r-1$. This giv…