As 0! and 1! = 1 ,
2! = 2,
3! = 6 and so on
Can we not have
-1! = -1
-2! = 2 = -1 X -2
-3! = -6 = -1 X -2 X -3
-4! = 24 = -1 X -2 X -3 X -4
-5! = -120 = -1 X -2 X -3 X -4 X -5
...and so on?
It would produce some sort of series but would they be of any use anywhere?

