co.combinatorics - Number of A Subset of Monomials

Let $S_i$ be the set of monic monomials $m in mathbb{Z}[x_1, dots,
x_k]$ which are divisible by $x_i$ but not $x_i^2$. If I am reading
your question correctly, you are looking for $|S_1 cup cdots cup
S_k|$.

Note that for $1 leq i_1 < dots < i_m leq k$, the intersection
$S_{i_1} cap cdots cap S_{i_m}$ is the set of monomials of degree
$n$ divisible by $x_{i_1} cdots x_{i_m}$ but not by $x^2_{i_1} cdots
x^2_{i_m}$. If $m < n$ and $m < k$, then there is a bijection between
$S_{i_1} cap cdots cap S_{i_m}$ and the set of monic monomials of
degree $n-m$ in $mathbb{Z}[x_1, dots, x_{k-m}]$. (If $m = n leq k$,
then the intersection has one element, $x_{i_1} cdots x_{i_m}$. In
any other case, the intersection is empty.) Hence, for $1 leq i_1 <
cdots < i_m leq k$,
$$|S_{i_1} cap cdots cap S_{i_m}| = begin{cases} left(matrix{n +
k - 2m -1 cr k - m - 1}right), & text{if $m < n$ and $m < k$} cr
1, & text{if $m = n leq k$} cr 0, & text{otherwise.}end{cases}$$

So, by the principle of inclusion-exclusion,
$$|S_1 cup cdots cup S_k| = sum_{m =1}^{min(n, k)-1} (-1)^{m-1}
left(matrix{k cr m}right)left(matrix{n + k - 2m - 1 cr k - m -
1}right) + (-1)^{n-1} left(matrix{kcr n}right)a,$$
where $$a = begin{cases} 1, & text{if $k geq n$} cr 0, &
text{otherwise.}end{cases}$$