An expansion for the number of partitions of an integer

Abstract

We consider p(n) the number of partitions of a natural number n, starting from an expression derived by Baez-Duarte in (Adv Math 125(1):114–120, 1997) by relating its generating function f(t) with the characteristic functions of a family of sums of independent random variables indexed by t. The asymptotic formula for p(n) follows then from a local central limit theorem as $$t\uparrow 1$$ suitably with $$n\rightarrow \infty $$. We take further that analysis and compute formulae for the terms that compose that expression, which accurately approximate them as $$t\uparrow 1$$. Those include the generating function f and the cumulants of the random variables. After developing an asymptotic series expansion for the integral term, we obtain an expansion for p(n) that can be simplified as follows: for each $$N>0$$, $$\begin{aligned} p(n)=\frac{2\pi ^2}{3\sqrt{3}} \,\frac{\text{ e }^{r_n}}{(1+2\,r_n)^2}\, \left( 1-\sum _{\ell = 1}^N\,\frac{D_\ell }{(1+2\,r_n)^\ell }+\,\mathcal R_{N+1}\,\right) . \end{aligned}$$The coefficients $$D_\ell $$ are positive and have simple expressions as finite sums of combinatorial numbers, $$r_n=\sqrt{\frac{2\pi ^2}{3}\,(n-\frac{1}{24})+\frac{1}{4}}$$ and the remainder satisfies $$n^{N/2}\,{\mathcal {R}}_{N+1} \rightarrow 0$$ as $$n\rightarrow \infty $$. The cumulants are given by series of rational functions and the approximate formulae obtained could be also of independent interest in other contexts.