Given an integer $n$ the number of partitions of $n$ into two colors can be represented as $$p_2(n)=\sum_{k=0}^n p(k)p(n-k)$$ where $p(k)$ counts the number of ordinary partitions of $k.$ What is the distribution of $$P(k)=\frac{p(k)p(n-k)}{p_2(n)}$$ as $n\to \infty.$

I feel as though this question has probably been addressed in the past but I am unaware of where in the literature. Does anyone know where I can find this?