Unfortunately, I don't have much detail to give here. But is the general idea to cancel out the constant obtained from taking the derivative.
For instance, say my function was $f(x)=f_o+f_1x+f_2x^2+...$
And if the expansion is centered around $x=0$...
And so forth. Is that where the factorial comes from?
It is quite clear for a polynomial, but what about a trig function such as $sin(x)$ other than using taylor's formula?