For any $q \in \mathbb{N}^{\ge 2}$, consider the polynomial $$ P(q) = \begin{cases} 1 + 2 \sum_{i = 1}^{\lceil q / 2 \rceil - 1} x^i + x^{q / 2} & \text{if $q$ is even}, \\ 1 + 2 \sum_{i = 1}^{\lceil q / 2 \rceil - 1} x^i & \text{else.} \end{cases} $$
The first few of them look as follows: \begin{align*} P(2) &= 1 + x, \\ P(3) &= 1 + 2x, \\ P(4) &= 1 + 2x + x^2, \\ P(5) &= 1 + 2x + 2x^2, \\ P(6) &= 1 + 2x + 2x^2 + x^3, \end{align*} and so on. For even $q$, they are obviously symmetric with respect to the coefficients. Consider now $P(q)^n$, for any $n \in \mathbb{N}$.
For $q \in \{2, 3, 4\}$, the $i$-th coefficient of $P(q)^n$ is $\binom{n}{i}$, $2^i \binom{n}{i}$ and $\binom{2n}{i}$, respectively. What I'm interested in is a general explicit form of the coefficients of $P(q)^n$ (i.e., for any $q$, not just 2, 3, 4).
Unfortunately, OEIS was not much of a help; it only contains the sequence for $q = 3$ (A013609). Does anybody have an idea? Has this been done before?
