Exact formulas for the partition function? 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.
 A: Jan Bruinier and Ken Ono just announced this.
A: This doesn't really answer the question, so perhaps it would be better as a comment, but alas, I don't have the necessary reputation.
Following up on Thomas Bloom's reference to the work of Bringmann and Ono, there is a paper of Folsom and Masri (Mathematische Annalen, available here: http://www.math.yale.edu/~alf8/Folsom-Masri-MathAnn07-10.pdf) which considers the main term one would get in an asymptotic formula arising from BO's Poincare series formula.  In particular, they also consider the problem of the error arising from truncating the infinite sum at $O(n^{1/2})$, obtaining power savings over the best known results of $O(n^{-1/2+\epsilon})$ if one truncates at $\lfloor \sqrt{n/6} \rfloor$.
A: I'm not sure whether this should be a comment or an answer: it is curiously missing from all the links above that the generating function for integer partitions satisfies a reasonably nice (order four, homogeneous of degree four) algebraic differential equation:
$$\begin{multline*}
4F^3 F'' + 5x F^3 F''' + x^2 F^3 F^{(\rm iv)} - 16F^2 F'^2  - 15x F^2 F' F'' + 20x^2 F^2 F' F'''\\
 - 39x^2 F^2 F''^2 + 10x F F'^3  + 12x^2 F F'^2 F'' + 6x^2 F'^4 = 0
\end{multline*}$$
There is actually also an order three differential equation, but it's not as nice.
According to Don Zagier [The 1-2-3 of modular forms, Section 5.1, Proposition 15]
already Ramanujan knew that every modular and every quasi-modular form
on $\Gamma_1$ satisﬁes a third order algebraic diﬀerential equation.  The equation above is found given the first 39 terms by 

guessADE([partition n for n in 0..39], homogeneous==4) 

from FriCAS in less than 0.01 seconds.
A: In answer to "What kind of exact formulas exist for the partition function", see http://arxiv.org/abs/1103.1585, eqs.(10), (11), and (12).
