We systematically investigate the expressions and congruences for both a one-parameter family $\left\{G_n\left(x\right)\right\}$ as well as a two-parameter family $\left\{G_n(r,m)\right\}$ of sequences.

A prime $p$ is said to be a Wolstenholme prime if it satisfies the congruence $\left(\genfrac{}{}{0pt}{}{2p-1}{p-1}\right)\equiv 1(mod{p}^4)$. For such a prime $p$, we establish an expression for $\left(\genfrac{}{}{0pt}{}{2p-1}{p-1}\right)(mod{p}^8)$ given in terms of the sums $R_i:={\sum }_{k=1}^{p-1}1/k^i$ ($i=1,2,3,4,5,6)$. Further, the expression in this congruence is reduced in terms of the sums $R_i$ ($i=1,3,4,5$). Using this congruence, we prove that for any Wolstenholme prime $p$ we have $$\left(\genfrac{}{}{0pt}{}{2p-1}{p-1}\right)\equiv 1-2p\sum _{k=1}^{p-1}\frac{1}{k}-2p^2\sum _{k=1}^{p-1}\frac{1}{k^2}(mod{p}^7).$$ Moreover, using a recent result of the author, we prove that a prime $p$ satisfying the above congruence must necessarily be a Wolstenholme prime. Furthermore, applying a technique...