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 $a_k$.