The alternating factorial is defined as the sum of consecutive factorials with alternating signs for a nonnegative
integer 
,
 |
(1)
|
It can be given in closed form as
![a(n)=(-1)^n[-1-eEi(-1)+(-1)^neE_(n+2)(1)Gamma(n+2)],](/images/equations/AlternatingFactorial/NumberedEquation2.svg) |
(2)
|
where 
is the exponential integral, 
is the En-function,
and 
is the gamma function.
The alternating factorial is implemented in the Wolfram Language as AlternatingFactorial[n].
A simple recurrence equation for 
is given by
 |
(3)
|
for 
,
with 
and 
.
For 
,
2, ..., the first few values are 1, 1, 5, 19, 101, 619, 4421, 35899, ... (OEIS A005165).
Alternating factorial primes are alternating factorials that are prime.
."
§B23, B43, and D25 in