A function that fills in the gaps between factorials seems useful. There is a simple formula which does just that.

That's almost what the Gamma function does, but not quite. The Gamma function calculates this:

Here's its common integral form:

It's like you're asking for the factorial of n, and the Gamma function says "Hey, I'll do you a favor and calculate the factorial of n - 1", and you're like "No, that's okay Gamma function, I really just want the factorial of n", and the Gamma function is like "No, I want to do this for you", and then it takes your number, subtracts one, computes the factorial, and hands you back the factorial of n - 1, all beaming and proud of itself for putting in the extra effort, and you have to smile and say thank you.