The result is $\boxed{\frac{1}{(1-x)^2}}$. This result can be obtained by the conclusion of # 4284.

Alternatively, it can also be obtained by taking derivative on the both sides of the following identity: $$1+x+x^2 + x^3 +\cdots = \frac{1}{1-x}$$

In general, if $f(x)$ is the generating function for the sequence $\{a_0, a_1, a_2, \cdots \}$, then $f'(x)$ is the generating function for the sequence $\{a_1, 2a_2, 3a_3, \cdots\}$. In another word, when a generating function is differentiated, each of its corresponding sequence will first be multiplied by its position index and then the whole sequence will be left shifted by one.