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.

中文翻译：

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关于涉及 Franel 数的两个同余

通过符号求和方法，我们为 $\pi^2$ 建立以下级数： \begin{align*} \sum_{k=1}^\infty \frac{H_k-2H_{2k}}{(-3)^ kk} = \frac{\pi^2}{18}, \end{align*} 其中 $H_k=\sum_{j=1}^k 1/j$。我们还推导出与这个系列相关的 $p$-adic 同余。作为应用，我们证明了两个涉及 Franel 数的同余，其中一个最初是由 Sun 猜想的。