  In this paper, we consider two particular binomial sums \begin{align*}
\sum_{k=0}^{n-1}(20k^2+8k+1){\binom{2k}{k}}^5 (-4096)^{n-k-1} \end{align*} and
\begin{align*} \sum_{k=0}^{n-1}(120k^2+34k+3){\binom{2k}{k}}^4\binom{4k}{2k}
65536^{n-k-1}, \end{align*} which are inspired by two series for
$\frac{1}{\pi^2}$ obtained by Guillera. We consider their divisibility
properties and prove that they are divisible by $2n^2 \binom{2n}{n}^2$ for all
integer $n\geq 2$. These divisibility properties are stronger than those
divisibility results found by He, who proved the above two sums are divisible
by $2n \binom{2n}{n}$ with the WZ-method.

Combinatorics;Number Theory