Some congruences involving inverse of binomial coefficients

Let $p$ be an odd prime number. In this paper, among other results, we establish some congruences involving inverse of binomial coefficients. These congruences are mainly determined modulo $p$, $p^{2}$, $p^{3}$ and $p^{4}$ in the $p$-integers ring in terms of Fermat quotients, harmonic numbers and Bernoulli numbers in a simple way. Furthermore, we extend an interesting theorem of E. Lehmer to the class of inverse binomial coefficients.