Finite differences of the partition function

Abstract:

From the Hardy-Ramanujan-Rademacher formula for $p(n)$—the number of unrestricted partitions of n, it is not difficult to deduce that there exists a least positive integer ${n_0}(r)$ such that ${V^r}p(n) \geqslant 0$ for each $n \geqslant {n_0}(r)$, where $Vp(n) = p(n) - p(n - 1)$ and ${V^r}p(n) = V\{ {V^{r - 1}}p(n)\}$. In this note, we give values of ${n_0}(r)$ for each $r \leqslant 10$ and conjecture that ${n_0}(r)/{r^3}\sim 1$.