Extending addictive factorial?

[waht]

If we define an addictive factorial for any integer n:

f(n) = n + (n-1) + (n-2) ... 0

1!+ = 1
2!+ = 2+1 = 3
3!+ = 3+2+1 = 6
4!+ = 4+3+2+1 = 10
5!+ = 15

is it possible to extend it to real or possibly complex numbers by analytic continuation?

just like the gamma function extends the factorial.

 

[Gerenuk]

Observe that
$$\sum_{k=1}^n k=\frac{n(n+1)}{2}$$
is your function.

 

[waht]

never mind, forgot about the binomial