I need to count the number of monomials of degree $n$ in $k$ variables, $x\_1,\ldots ,x\_k$, that contain at least one variable with a power of 1. The monomials need not include all the variables. Their powers just need to some to $n$ and they must be divisible by $x\_i$, but not $x\_i^2$, for some $i$.



Number of A Subset of Monomials
Another formula (almost without alternating signs) can be obtained as a variation of the comment of David Speyer. Namely, for each $S\subset\{1,\ldots,k\}$ we can consider the set of all monomials that depend precisely on all $x\_k$ with $k\in S$ and \emph{do not satisfy the property we are studying}. Such a monomial is divisible by $\prod\_{k\in S}x\_k^2$, so the number of such monomials is $\binom{|S|+n-2|S|-1}{|S|-1}$. The number of choices for $S$ is $\binom{k}{|S|}$, so altogether the number of ``unwanted'' monomials
is $$\sum\_{s=1}^{k}\binom{k}{s}\binom{n-s-1}{s-1},$$ 
and the number of monomials you want to compute is
$$\binom{k+n-1}{k-1}-\sum\_{s=1}^{k}\binom{k}{s}\binom{n-s-1}{s-1}.$$ 


