A partition of $n$ is a weakly decreasing tuple of numbers $(\lambda_1,\lambda_2,\lambda_3,....\lambda_k)$ whose sum is $n.$ A natural problem studied is counting partitions whose summands $\lambda_i$ all like in a set $S\subset \mathbb{N}$ and we denote this number $p_S(n)$.

If $S$ is ``dense" enough, one can usually calculate an asymptotic formula that looks like $$p_S(n) = Kn^\beta\exp\left(c\sqrt{n}\right)\left(1+ \frac{c_1}{\sqrt{n}}+ \frac{c_2}{n}+\frac{c_3}{n^\frac{3}{2}} \dots \right).$$

My question is when $S=\mathbb{N}\setminus [0,1,2,\dots c]$ does such a formula already exist in the literature and where can I find it?