  The Raney numbers $R_{p,r}(k)$ are a two-parameter generalization of the
Catalan numbers. In this paper, we obtain a recurrence relation for the Raney
numbers which is a generalization of the recurrence relation for the Catalan
numbers. Using this recurrence relation, we confirm a conjecture posed by
Amdeberhan concerning the enumeration of $(s,s+1)$-core partitions $\lambda$
with parts that are multiples of $p$. We then give a new combinatorial
interpretation for the Raney numbers $R_{p+1,r+1}(k)$ with $0\leq r<p$ in terms
of $(kp+r,kp+r+1)$-core partitions $\lambda$ with parts that are multiples of
$p$.
