What is $x^2-1$ applied n times

Question

For the function $F(x)=x^2-1$. How do I write $F^n(x)$ ($F$ applied $n$ times) in terms of $x$?

Answer 1

A somewhat trivial way would be $F^n(x) = x^2-1$ if $n=1$ and $F^n(x)=(F^{n-1}(x))^2-1$ otherwise.

Presumably you want a closed form, though, and it is probably pretty obvious that in the closed form the leading coefficient is 1, the degree is $2n$, and the constant term is $-1$ if $n$ is odd and 0 if $n$ is even. Perhaps if you write out large examples you'll see a pattern for intermediate terms, perhaps expressible by some kind of manipulation of the choose function.

Answer 2

This sequence is analyzed in this OEIS sequence: http://oeis.org/A158984. Note that the solution given there is recursive as well.