I am looking for a closed form for the sum below
$$ \sum_{k=0}^{r} \frac{1}{2-\delta_{k, r-k}} \binom{n}{k} \binom{n}{r-k} $$
where $\delta_{k, r-k}$ is the Kronecker delta. I'd appreciate any helps in finding a closed-form solution for this sum.
To my understanding, this is basically equivalent to finding a closed-form solution to the following problem:
Given a set $S$ with size $|S|=n$, what is the number of two-element subsets of $\mathcal{P}(S)$, $\{A, B\} \in \mathcal{P}(S)$ such that $|A|+|B| = r$ for a fixed $r$.
I tried to use Wolfram Alpha, but it is given me this for the case where we count ordered pairs $(A, B)$ instead of sets (i.e. the sum if we ignore $\frac{1}{2-\delta_{k, r-k}}$), which contains factorial of a negative integer(!!) (and I don't know how to specify in Wolfram that $n>r>0$ and both are integers):
$$ \frac{(-1)^r (-2n+r-1)!}{(-2n-1)!r!} $$