The term "left factorial" is sometimes used to refer to the subfactorial 
, whose first few values for 
, 2, ... are 0, 1, 2, 9, 44, 265, 1854, ... (OEIS A000166).
The same term and notation are also applied, for a nonnegative integer 
, to the factorial sum
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
is the gamma function, 
is the upper incomplete
gamma function on its principal branch, 
is the exponential
integral, and 
is the En-function.
For 
,
1, ..., the first few values are given by 0, 1, 2, 4, 10, 34, 154, 874, ... (OEIS
A003422). The left factorial is always even
for 
.
The related factorial sum 
has values 1, 3, 9, 33, 153, 873,
5913, ... for 
,
2, ... (OEIS A007489).
is prime
for 
,
4, 5, 8, 9, 10, 11, 30, 76, 163, 271, 273, 354, 721, 1796, 3733, 4769, 9316, 12221,
... (OEIS A100614), the last of which was found
by E. W. Weisstein (Oct. 19, 2006).
." Math. Balkanica 1, 147-153, 1971.Kurepa,
D. "Left Factorial Function in Complex Domain." Math. Balkanica 3,
297-307, 1973.Sloane, N. J. A. Sequences