We prove two supercongruences for sums of the Apéry-like numbers:

$$\begin{aligned} T_n=\sum _{k=0}^n{n\atopwithdelims ()k}^2{2k\atopwithdelims ()n}^2, \end{aligned}$$

which was first introduced by Almkvist and Zudilin. These results confirm two conjectural supercongruences due to Sun. Our proof relies on symbolic summation method and Sun’s transformation formula.

中文翻译：

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关于类 Apéry 数和的两个超同余

我们证明了类似 Apéry 的数之和的两个超同余：

$$\begin{aligned} T_n=\sum _{k=0}^n{n\atopwithdelims ()k}^2{2k\atopwithdelims ()n}^2, \end{aligned}$$

这是由 Almkvist 和 Zudilin 首次引入的。这些结果证实了由 Sun 引起的两个推测性超同余。我们的证明依赖于符号求和方法和 Sun 的变换公式。