Short proof of Rademacher's formula for partitions (Q2316343)

The authors provide a short proof of Rademacher's formula for the partition functions \(p_r(n)\), given by \[\sum_{n=0}^\infty p_r(n) q^n = \prod_{n=1}^\infty \frac{1}{(1-q^n)^r},\qquad 1\leqslant r\leqslant 24.\] This proof uses only the Fourier expansion of Poincaré series and the fact that any weight \(2\) modular form has constant term \(0\).