The complementary Bell numbers, also called the Uppuluri-Carpenter numbers,
 |
(1)
|
where 
is a Stirling number of the second
kind, are defined by analogy with the Bell numbers
 |
(2)
|
They are given by
 |
(3)
|
where 
is a Bell polynomial.
For 
,
1, ..., the first few are 1, 
, 0, 1, 1, 
, 
, 
, 50, 267, 413, ... (OEIS A000587).
They have generating function
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
They have the series representation
 |
(7)
|
They are prime (in absolute value) for 
, 36, 723, ... (OEIS A118018),
corresponding to the prime numbers 2, 1454252568471818731501051, ... (OEIS A118019),
with no others for 
(E. W. Weisstein, Mar. 21, 2009).
and 
."
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