On Divisibility of Convolutions of Central Binomial Coefficients

Recently, Z. Sun proved that \[ 2(2m+1)\binom{2m}{m} \mid \binom{6m}{3m}\binom{3m}{m} \] for $m\in\mathbb{Z}_{>0}$. In this paper, we consider a generalization of this result by defining \[ b_{n,k}=\frac{2^{k}\, (n+2k-2)!!}{((n-2)!!\, k!}. \] In this notation, Sun's result may be expressed as $2\, (2m+1) \mid b_{(2m+1),(2m+1)-1}$ for $m\in\mathbb{Z}_{>0}$. In this paper, we prove that \[ 2n \mid b_{n,un\pm 2^{r}} \] for $n\in\mathbb{Z}_{>0}$ and $u,r\in\mathbb{Z}_{\geq 0}$ with $un \pm 2^{r} > 0$. In addition, we prove a type of converse. Namely, fix $k\in\mathbb{Z}$ and $u\in \mathbb{Z}_{≥0}$ with $u>0$ if $k0}$ with $un+k>0$, then there exists a unique $r \in \mathbb{Z}_{≥0}$ so that either $k=2^{r} $ or $k=-2^{r}$.