What is the $C$ constant in this generating function? (probability)

Question

Let

\begin{equation*} G(x)= C \frac{4x^4+x^5+1}{16-8x-4x^2}. \end{equation*}

How am I supposed to calculate $C$? Out of $50$ experiments how many $0$'s do I get?

$16-8x-4x^2$ can be written as: $-(2x+2)^2 +20$, but it doesn't look like the generating function of the binomial distribution.

Thank you in advance!

Answer 1

Hints:

1. To quote Wikipedia:

The normalization of the probability density function can be expressed in terms of the generating function by $$\operatorname{E}(1)=G(1^-)=\sum_{i=0}^\infty f(i)=1.$$

2. $P(X=0)=G(0)$