2020? But it's still 2015!

Algebra Level 4

$\large f(x+1) = (5^{2^{x-1}} + 1)(x+1)^{2}f(x)$

Suppose we define a function of $x$ as described above for $x$ is a non-negative integer with $f(0)=\sqrt{5} - 1$ .

If the value of $f(2020)$ can be written in form $(5^{p} - 1)(q!)^{2}$ , for positive integers $p$ and $q$ , find $\frac{2^{q}}{p}$ .