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

Abstract: In this note, we show that $S(n,r):=\sum_{k=0}^{n} \binom{n}{k}\frac{k}{k+r}$ 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 gives a partial answer to a conjecture of [3].