The Raney numbers   R p,r (k)       R    p  ,  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   λ     λ        with parts that are multiples of   p     p       . As a corollary, we give a new combinatorial interpretation for the Raney numbers   R p+1,r+1 (k)       R    p  +  1  ,  r  +  1     (  k  )         with   0≤r     0  ≤  r    p        in terms of   (kp+r,kp+r+1)      (  k  p  +  r  ,  k  p  +  r  +  1  )        -core partitions   λ     λ        with parts that are multiples of   p     p       .
['Robin D. P. Zhou']
The Raney Numbers and $(s,s+1)$-Core Partitions