Euler's pentagonal number theorem relates the partition function $p(n)$ (number of ways to partition $n$) via the recurrence: $p(n) = \sum_{k \neq 0} (-1)^{k+1} p\left(n - \frac{k(3k-1)}{2}\right)$ Given that $p(0) = 1, p(1) = 1, p(2) = 2, p(3) = 3, p(4) = 5$, compute $p(5)$ using the first few terms of this recurrence.