Equation to approximate the Partition Function

Question

In this Numberphile video, the partition function is discussed. At some point in the video, an approximation function is briefly talked about: It is mentioned that the approximation equation has additional terms that can make it more accurate and in using four terms the approximation it will result in an answer accurate to the nearest integer for n>200. The video didn't link anything to this extended equation and I looked online briefly but couldn't find anything for this equation. Does anyone know where I could find it?

Answer 1

We find in section VIII.22 of Analytic Combinatorics by P. Flajolet and R. Sedgewick:

The number of integer partitions $p(n)$ satisfies the Hardy-Ramanujan-Rademacher expansion which is the exact formula \begin{align*} p(n)=\frac{1}{\pi\sqrt{2}}\sum_{k=1}^\infty A_k(n)\sqrt{k}\frac{d}{dn}\frac{\sinh \left(\frac{\pi}{k} \sqrt{\frac{2}{3}\left(n-\frac{1}{24}\right)}\right)}{\sqrt{n-\frac{1}{24}}} \end{align*} where \begin{align*} A_k(n)=\sum_{{h \mod k}\atop{gcd(h,k)=1}}\omega_{h,k}e^{-2 i\pi h/k} \end{align*} and $\omega_{h,k}$ is the $24$th root of unity, $\omega_{h,k}=\exp(\pi i s(h,k))$ and \begin{align*} s_{h,k}=\sum_{\mu=1}^{k-1}\left\{\left\{\frac{\mu}{k}\right\}\right\} \left\{\left\{\frac{h\mu}{k}\right\}\right\} \end{align*} is known as a Dedekind sum, with $\{\{x\}\}=x-\lfloor x\rfloor -\frac{1}{2}$.

Some information is also stated in Wiki.