How can I determine the sequence generated by a generating function? For example in my textbook it says that the sequence for the generating function
$$f(x) = (2x-3)^3$$
is
$$-27,54,-36,8,0,0,0,\dots$$
I can see that plugging $0$ into the function gives me $-27$, but this doesn't hold true for $1$ and $54$. How do I get this sequence or the correct sequence for another generating function like
$$f(x) = \frac{x^3}{1-x^2}\;?$$

For the first example, these are the cofficents of the polynomial.
For the second example, $1/(1-x^2)=1+x^2+x^4+.....$, therefore $x^3/(1-x^2)=x^3+x^5+x^7+....$, thus the generating sequence is : 0,0,0,1,0,1,0,1,0,...
