Binomial Coefficient Question In the expression
$((. . . ((x − 2)^2 − 2)^2 − 2)^2 − · · · − 2)^2 − 2)^2$
,
there are $k$ pairs of parentheses, where k is a positive integer. In terms of $k$, find the coefficient of $x^2$
after expanding and collecting the terms
So far I have attempted to look for a recursion which would allow me to solve the problem. However, I cannot seem to find a relationship between the input of $k$ and the coefficient of the $x^2$ term. Can anybody help?

Hint. Let $F_k(x)$ the the function at the $k$-th step, then 
$$F_k(x)=a_kx^2+b_kx+4+o(x^2)$$
Then $F_1=x^2-4x+4$ implies that $a_1=1$ and $b_1=-4$.
Moreover
\begin{align*}a_{k+1}x^2+b_{k+1}x+4+o(x^2)&=F_{k+1}(x)=(F_k(x)-2)^2\\
&=(a_kx^2+b_kx+4+o(x^2)-2)^2\\
&=(4a_k+b_k^2)x^2+4b_kx+4+o(x^2).
\end{align*}
Therefore $b_{k+1}=4b_k=-4^{k+1}$ and 
$$a_{k+1}=4a_k+b_k^2=4a_k+16^k.$$
It remains to solve the linear recurrence for $a_k$.
