The Riordan number 
,
for a nonnegative integer 
, is the number of set partitions
of 
points placed on a circle
such that every set block contains at least two points
and the convex hulls of distinct blocks are disjoint
(Bernhart 1999). The first few Riordan numbers are 1, 0, 1, 1, 3, 6, 15, 36, 91,
232, 603, 1585, ... (OEIS A005043). They are
also called ring numbers (Bernhart 1999).
The Riordan numbers satisfy the recurrence relation
 |
(1)
|
for 
,
with 
and 
.
Their generating function is
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
It obeys the functional equation
 |
(5)
|
If 
is the 
th
Motzkin number, then
 |
(6)
|
Equivalently, 
counts ordered rooted trees with 
edges in which no vertex
has exactly one child (Bernhart 1999).
