Cross posting from MSE.

**Definition:**

For any natural number $n\ge 3$, define the polynomial $P_{n}\left(x_1,x_2,...,x_{n-1},x_{n} \right)$, with indeterminates $x_{i}$, where $i\in\{1,2,...,n-1,n\}$, by the following recursive rule:
$$
\begin{align}
P_{n}\left(x_1,x_2,...,x_{n-1},x_{n} \right) \equiv
\begin{cases}
\left(x_1+x_2+x_3\right)^2-2 \left(x_{1}^2+x_{2}^2+x_{3}^2 \right), & \text{if } n=3 \\\
\mathrm{Res}_{\mathbf{x}}\left(P_{3}\left(x_1,x_2,\mathbf{x} \right),P_{n-1}\left(x_3,x_4,...,x_{n},\mathbf{x} \right) \right), & \text{if } 4 \le n
\end{cases}
\end{align}
$$

Observe that if $4 \le n$ then $P_{n}\left(x_1,x_2,...,x_n \right)= \mathrm{Res}_{\mathbf{x}}\left(P_{n-1}\left(x_1,x_2,...,x_{n-2},\mathbf{x} \right),P_{3}\left(x_{n-1},x_n,\mathbf{x} \right) \right)$.

**Example #1:**

If $n=4$ then
$$
\begin{align}
P_{4}\left(x_1,x_2,x_3,x_4 \right)=\mathrm{Res}_{\mathbf{x}}\left(P_{3}\left(x_1,x_2,\mathbf{x} \right),P_{3}\left(x_3,x_4,\mathbf{x} \right) \right)=\left(\left(x_1+x_2+x_3+x_4\right)^2-2 \left(x_{1}^2+x_{2}^2+x_{3}^2+x_{4}^2 \right)\right)^2-64x_{1}x_{2}x_{3}x_{4}
\end{align}
$$

**Example #2:**

If $n=5$ then
$$
\begin{align}
P_{5}\left(x_1,x_2,x_3,x_4,x_5 \right)=\mathrm{Res}_{\mathbf{x}}\left(P_{3}\left(x_1,x_2,\mathbf{x} \right),P_{4}\left(x_3,x_4,x_5,\mathbf{x} \right) \right)=\left(\left(\left(x_1+x_2+x_3+x_4+x_5\right)^2-2 \left(x_{1}^2+x_{2}^2+x_{3}^2+x_{4}^2+x_{5}^2 \right)\right)^2-64\left( x_{1}x_{2}x_{3}x_{4}+x_{1}x_{2}x_{3}x_{5}+x_{1}x_{2}x_{4}x_{5}+x_{1}x_{3}x_{4}x_{5}+x_{2}x_{3}x_{4}x_{5}\right) \right)^2-2048x_{1}x_{2}x_{3}x_{4}x_{5}\left(8\left(x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+x_{1}x_{2}x_{5}+x_{1}x_{3}x_{4}+x_{1}x_{3}x_{5}+x_{1}x_{4}x_{5}+x_{2}x_{3}x_{4}+x_{2}x_{3}x_{5}+x_{2}x_{4}x_{5}+x_{3}x_{4}x_{5}\right) -\left(x_1+x_2+x_3+x_4+x_5 \right)\left( \left(x_1+x_2+x_3+x_4+x_5\right)^2-2 \left(x_{1}^2+x_{2}^2+x_{3}^2+x_{4}^2+x_{5}^2 \right)\right)\right)
\end{align}
$$

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".

**Possible lead #1:**

The object at hand (might, still) "screams" partition-related symmetric-polynomial objects, like the Schur Polynomials and their generalizations. Unfortunately, playing around with those hasn't produced a hopeful pattern for a general formula, yet.

**Possible lead #2:**

Even though $P_n$ is a symmetric polynomial, so it has a canonical representation in *Elementary Symmetric Polynomials*, and some powerful tools for manipulation, it just seems to me, from playing around, that the complexity of $P_{n+1}$ explodes when compared to $P_n$ when they are represented with symmetric polynomials. Another direction might have something to do with identities involving powers of sums of squares, like those apperaing on this wonderful page.

**Major Edit (definition changed):**

Changed the definition of $P_n$ to an equivalent, more simple one, so one doesn't have to check whether $P_n$ is well defined. The original definition can still be seen on the MSE question (link at the top of this question).