Generation function for recurrence Could you tell me how I can find the generation function for recurrence $\sum_{n = 0}^{ \infty} n a_n t^n$ if I know $A(t)$ - generation function for $a_0, a_1, a_2 \dots$  . 
Thanks

You know that:
$$\sum_{n=0}^{\infty}a_nt^n=A(t)$$
Derive both sides (left side term by term).You get:
$$\sum_{n=0}^{\infty}na_nt^{n-1}=A'(t)$$
Now multiply both sides by $t$:
$$\sum_{n=0}^{\infty}na_nt^{n}=tA'(t)$$
