If the ordinary generating function of a sequence $\{a_n\}_{n=0}^{\infty}$ is $\frac{1+z}{(1-z)^3}$, then $a_3 - a_0$ is equal to ________.

The ordinary generating function is given by:

Using the standard binomial series expansion:

For $k = 3$:

Multiplying by $(1+z)$:

Hence, the general term $a_n$ (for $n \ge 1$) is:

Calculating the required values:

• For $n = 0$: $a_0 = (0+1)^2 = 1$

• For $n = 3$: $a_3 = (3+1)^2 = 16$

Therefore: