Congruences involving $$g_n(x)=sum limits _{k=0}^nleft( {begin{array}{c}n kend{array}}right) ^2left( {begin{array}{c}2k kend{array}}right) x^k$$

Define (g_n(x)=sum _{k=0}^nleft( {begin{array}{c}n kend{array}}right) ^2left( {begin{array}{c}2k kend{array}}right) x^k) for (n=0,1,2,ldots ). Those numbers (g_n=g_n(1)) are closely related to Apéry numbers and Franel numbers. In this paper we establish some fundamental congruences involving (g_n(x)). For example, for any prime (p>5) we have

$$begin{aligned} sum _{k=1}^{p-1}frac{g_k(-1)}{k}equiv 0pmod {p^2}quad text {and}quad sum _{k=1}^{p-1}frac{g_k(-1)}{k^2}equiv 0pmod p. end{aligned}$$

This is similar to Wolstenholme’s classical congruences

$$begin{aligned} sum _{k=1}^{p-1}frac{1}{k}equiv 0pmod {p^2}quad text {and}quad sum _{k=1}^{p-1}frac{1}{k^2}equiv 0pmod p end{aligned}$$

for any prime (p>3).