I want to find the generating function for a_{n}=n(n-1).

$\displaystyle \sum_{n=0}^{\infty }n(n-1)\frac{x^n}{n!}$

$\displaystyle =\sum_{n=0}^{\infty }\frac{x^n}{(n-2)!}$

$\displaystyle =x^2\sum_{n=0}^{\infty }\frac{x^{n-2}}{(n-2)!}$

$\displaystyle =x^2\sum_{k=-2}^{\infty }\frac{x^{k}}{k!}$

$\displaystyle =x^2\sum_{k=0}^{\infty }\frac{x^{k}}{k!}$

$\displaystyle =x^2e^x$

Am I justified in the second to last step, ignoring the summands for k=-1 and k=-2?

If so, is there a reason why I can ignore the first two terms?