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 up arrow 1We take further that analysis and compute formulae for the terms that compose that expression, which accurately approximate them as t up arrow 1Those 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 p(n)=2 pi 233ern(1+2rn)2mml:mfenced close=")" open="("1- n-ary sumation l=1NDl(1+2rn)l+RN+1.$$\begin{aligned} p(n)=\frac{2\pi ^>2}{3\sqrt{3}} \,\frac{\text{ e }^>{r_n}}{(1+2\,r_n)^>2}\, \left( 1-\sum _{\ell = 1}^>N\,\frac{D_\ell }{(1+2\,r_n)^>\ell }+\,\mathcal R_{N+1}\,\right) . \end{aligned}$$\end{document}The coefficients Dl are positive and have simple expressions as finite sums of combinatorial numbers, rn=2 pi 23(n-124)+14 and the remainder satisfies nN/2RN+1 -> 0 The cumulants are given by series of rational functions and the approximate formulae obtained could be also of independent interest in other contexts.