On Wolstenholme's theorem and its converse

For any positive integer n, let w_n = ((2 n - 1; n - 1)) = frac(1, 2) ((2 n; n)). Wolstenholme proved that if p is a prime ≥5, then w_p ≡ 1 (mod p³). The converse of Wolstenholme's theorem, which has been conjectured to be true, remains an open problem. In this article, we establish several relations and congruences satisfied by the numbers w_n, and we deduce that this converse holds for many infinite families of composite integers n. In passing, we obtain a number of congruences satisfied by certain classes of binomial coefficients, and involving the Bernoulli numbers.