2020? But it's still 2015!

Algebra Level 5

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

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 \(\frac{2^{q}}{p}\).

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