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Define $g_n(x)=\sum_{k=0}^n\binom nk^2\binom{2k}kx^k$ for $n=0,1,2,...$.Those numbers $g_n=g_n(1)$ are closely related to Ap\'ery numbers and Franelnumbers. In this paper we establish some fundamental congruences involving$g_n(x)$. For example, for any prime $p>5$ we have$$\sum_{k=1}^{p-1}\frac{g_k(-1)}{k}\equiv0\pmod{p^2}\quad{and}\quad\sum_{k=1}^{p-1}\frac{g_k(-1)}{k^2}\equiv 0\pmod p.$$This is similar to Wolstenholme's classical congruences$$\sum_{k=1}^{p-1}\frac1k\equiv0\pmod{p^2}\quad{and}\quad\sum_{k=1}^{p-1}\frac{1}{k^2}\equiv0\pmodp$$ for any prime $p>3$.

Congruences involving $g_n(x)=\sum_{k=0}^n\binom nk^2\binom{2k}kx^k$