On the Numbers of Ramanujan

In Chapter 3 of his second notebook, Ramanujan defined numbers a (n, k) such that a (2, 0) = 1 and for n ≥ 2, a (n + 1, k) = (n − 1) a (n, k − 1) + (2n − 1 − k) a (n, k), where a(n, k) = 0 when kn − 2. These numbers are expressed in terms of Stirling numbers of the first kind and associated Stirling numbers of the second kind, and satisfy certain divisibility properties. In this paper, we obtain further properties for a(n, k), including a new characterization of prime numbers. We also derive several congruences for the numbers of Ramanujan modulo p2, some of which lead to new conditions for a prime number to be a Wilson prime.