  Via symbolic summation method, we establish the following series for $\pi^2$:
\begin{align*} \sum_{k=1}^\infty \frac{H_k-2H_{2k}}{(-3)^k k} =
\frac{\pi^2}{18}, \end{align*} where $H_k=\sum_{j=1}^k 1/j$. We also derive a
$p$-adic congruence related to this series. As an application, we prove two
congruences involving Franel numbers, one of which was originally conjectured
by Sun.
