Suppose that the characteristic of the field $\mathbb{F}$ is not $2$.
Definition:
For any natural number $n$, such that $3 \le n$, and for any $x_{i} \in \mathbb{F}$, where $i\in\{1,2,...,n-1,n\} $, define the polynomial $P_{n}\left(x_1,x_2,...,x_{n-1},x_{n} \right)$ by the following recursive rule:
$$
\begin{align}
P_{n}\left(x_1,x_2,...,x_{n-1},x_{n} \right) \equiv
\begin{cases}
\left(x_1+x_2+x_3\right)^2-2 \left(x_{1}^2+x_{2}^2+x_{3}^2 \right), & \text{if } n=3 \\\
Res_{\mathbf{x}}\left(P_{n_0}\left(x_1,x_2,...,x_{n_0-1},\mathbf{x} \right),P_{n_1}\left(x_{\left(n_0-1\right)+1},x_{\left(n_0-1\right)+2},...,x_{\left(n_0-1\right)+\left(n_1-1\right)},\mathbf{x} \right) \right), & \text{if } 4 \le n
\end{cases}
\end{align}
$$
where $n_{0},n_{1}$ are natural numbers such that $3 \le n_{0},n_{1}$ and $n_{0}+n_{1}=n+2$.
Please convince yourself that this is well defined due to the properties of the resultant and the fact that $P_{3}\left(x_1,x_2,x_3 \right)$ is a symmetric polynomial.
Example #1:
If $n=4$ then denote $n_0 \equiv 3$ and $n_1 \equiv 3$ to obtain that
$$
\begin{align}
P_{4}\left(x_1,x_2,x_3,x_4 \right)=Res_{\mathbf{x}}\left(P_3\left(x_1,x_2,\mathbf{x} \right),P_{3}\left(x_3,x_4,\mathbf{x} \right) \right)=\left(\left(x_1+x_2+x_3+x_4\right)^2-2 \left(x_{1}^2+x_{2}^2+x_{3}^2+x_{4}^2 \right)\right)^2-64x_{1}x_{2}x_{3}x_{4}
\end{align}
$$
Example #2:
If $n=5$ then denote $n_0 \equiv 4$ and $n_1 \equiv 3$ to obtain that
$$
\begin{align}
P_{5}\left(x_1,x_2,x_3,x_4,x_5 \right)=Res_{\mathbf{x}}\left(P_4\left(x_1,x_2,x_3,\mathbf{x} \right),P_{3}\left(x_4,x_5,\mathbf{x} \right) \right)=\left(\left(\left(x_1+x_2+x_3+x_4+x_5\right)^2-2 \left(x_{1}^2+x_{2}^2+x_{3}^2+x_{4}^2+x_{5}^2 \right)\right)^2-64\left( x_{1}x_{2}x_{3}x_{4}+x_{1}x_{2}x_{3}x_{5}+x_{1}x_{2}x_{4}x_{5}+x_{1}x_{3}x_{4}x_{5}+x_{2}x_{3}x_{4}x_{5}\right) \right)^2-2048x_{1}x_{2}x_{3}x_{4}x_{5}\left(8\left(x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+x_{1}x_{2}x_{5}+x_{1}x_{3}x_{4}+x_{1}x_{3}x_{5}+x_{1}x_{4}x_{5}+x_{2}x_{3}x_{4}+x_{2}x_{3}x_{5}+x_{2}x_{4}x_{5}+x_{3}x_{4}x_{5}\right) -\left(x_1+x_2+x_3+x_4+x_5 \right)\left( \left(x_1+x_2+x_3+x_4+x_5\right)^2-2 \left(x_{1}^2+x_{2}^2+x_{3}^2+x_{4}^2+x_{5}^2 \right)\right)\right)
\end{align}
$$
As you can see, $P_n$ becomes quite complicated fairly quickly.
Question:
I want to find a closed form expression for $P_n$. I doubt that this is feasible, so I also look for alternative ways to compute $P_n$ which may be quicker and more efficient. I would also like to know if there are papers or theories which deal with this sort of polynomial objects. Keeping this in mind, I realize that this question may be regarded as "soft".