The complementary Bell number 
, also called an Uppuluri-Carpenter number, is defined
for a nonnegative integer 
by
 |
(1)
|
where 
is a Stirling number of the second
kind, 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).
Their exponential generating function 
is
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
They have the series representation
 |
(7)
|
where 
.
The reported indices of prime or probable
prime absolute values begin 
, 36, 723, ... (OEIS A118018).
The corresponding values begin 2, 1454252568471818731501051, ... (OEIS A118019).
The value at 
is reported as a probable prime. A search was reported
to find no other candidates for 
(E. W. Weisstein, pers. comm., Mar. 21,
2009).
and 
."
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and Enumeration Problems of Combinatorial Analysis (Ed. G. P. Egoryčev
and M. L. Platonov). Krasnoyarsk, Soviet Union: Krasnojarsk. Gos. Univ.,
pp. 117-124, 1976.Layman, J. W. and Prather, C. L. "Generalized
Bell Numbers and Zeros of Successive Derivatives of an Entire Function." J.
Math. Anal. Appl. 96, 42-51, 1983. 
." Fib. Quart. 7, 437-448, 1969.Yang,
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