I am curious, what kind of exact formulas exist for the partition function $p(n)$?

I seem to remember an exact formula along the lines $p(n) = \sum_k f(n, k)$, where $f(n, k)$ was some extremely messy transcendental function, and the approximation was so good that for large $n$ one could simply take the $k = 1$ term and truncate this to the nearest integer to get an exact formula.

Reviewing the literature, it seems that I misremembered Rademacher's exact formula, which *is* of the above type but which requires more than one term. I am curious if there are other exact formulas, particularly of the type I mentioned?

Also, if I am indeed wrong and no such formula has been proved, is some good reason why it would be naive to expect one?

Thanks.