The Raney numbers and (s,s+1)(s,s+1)-core partitions

The Raney numbers Rp,r(k)Rp,r(k) are a two-parameter generalization of the Catalan numbers. In this paper, we give a combinatorial proof for a recurrence relation of the Raney numbers in terms of coral diagrams. Using this recurrence relation, we confirm a conjecture posed by Amdeberhan concerning the enumeration of (s,s+1)(s,s+1)-core partitions λλ with parts that are multiples of pp. As a corollary, we give a new combinatorial interpretation for the Raney numbers Rp+1,r+1(k)Rp+1,r+1(k) with 0≤r