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.