How to prove this recursion for these Taylor coefficients?

Question

Consider the Taylor series

$$ f(x) = \frac{1 - 2x - \sqrt{1 - 8 x + 8 x^2}}{2x(1-x)} = f_0 + f_1 x + f_2 x^2 + ... $$

It appears that the Taylor coëfficiënts start to follow the recursion

$$ 0 = (n+1)f_n + 3(1 - 3n) f_{n-1} + 4(4n -5) f_{n-2} + 8(2-n) f_{n-3} $$

(Hope i made no typo.)

How to prove this efficiently ?

Answer 1

First of all, notice that $f$ is undefined at $x=0$ and has no derivatives there. But this is a removable singularity, writing $f$ in the equivalent form $$f(x)=\frac{2}{1-2x+\sqrt{1-8x+8x^2}}$$ allows us to avoid this.

Now the first few coefficients are $f_0=1$, $f_1=3$, $f_2=11$ and $f_3=47$. Plugging these into the formula with $n=3$ starts the induction.

Can you continue now?