In a standard $52$ card deck of playing cards, a set of $4$ cards is a Badugi if it contains $4$ cards with no pairs and no two cards of the same suit.

Let $f(n)$ be the number of ways to choose $n$ cards with a $4$ card subset that is a Badugi. For example, there are $2598960$ ways to choose five cards from a standard $52$ card deck, of which $514800$ contain a $4$ card subset that is a Badugi, so $f(5)=514800$.

Find $\sum f(n)$ for $4\le n\le 13$.