\[p\left(n\right)=\sum_{k=1}^{\infty}(-1)^{k+1}\left(p\left(n-\omega(k)\right)+p%
\left(n-\omega(-k)\right)\right)=p\left(n-1\right)+p\left(n-2\right)-p\left(n-%
5\right)-p\left(n-7\right)+\cdots,\]