Consider the $n$-th power of the elementary symmetric polynomial $e_2(x_1,\ldots,x_n)$:
$$e_2^n(x_1,\ldots,x_n)=\left( \sum_{1 \le i \lt j \le n}{x_ix_j} \right)^n$$
and the number of classes of terms isomorphic up to a permutation of the $x_1,\ldots,x_n$ variables. In other words, the class is uniquely identified by the multiset of the exponents of the variables in the term (e.g. for $7x_1^3x_2^3x_3^2x_4$ it is $\{1,2,3,3\}$). For example:
$$e_2^2(x_1,x_2)=x_1^2x_2^2$$
and there is only one class of terms. For $n=3$:
$$e_3^3(x_1,x_2,x_3)=x_2^3 x_1^3 + x_3^3 x_1^3 + 3 x_2 x_3^2 x_1^3 + 3 x_2^2 x_3 x_1^3 + 3 x_2 x_3^3 x_1^2 + 6 x_2^2 x_3^2 x_1^2 + 3 x_2^3 x_3 x_1^2 + 3 x_2^2 x_3^3 x_1 + 3 x_2^3 x_3^2 x_1 + x_2^3 x_3^3$$
and there are $3$ different classes of terms, because for example $3 x_2 x_3^2 x_1^3$ is equivalent to $3 x_2 x_3^3 x_1^2$ and $x_2^3 x_1^3$ to $x_3^3 x_1^3$.
With the help of Wolfram Alpha, I have computed other two terms of the sequence, which is then: $1,3,8,18$ for $n=2,3,4,5$. I am not able to go further without writing a program (and I fear it doesn't have a reasonable computation time for $n=6$), and up to now I didn't find anything fit among the OEIS sequences that include $1,3,8,18$.
Any hint for deriving a formula or at least an asymptotic approximation?