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.