Suppose you have a series $\{A_n\}$ with the following recurrence relation: $$A_{n+1} = \sum_{\lambda(n)}\prod_{i=1}^{|\lambda|}A_{\lambda_i}$$ where $\lambda(n)$ is an integer partition of $n$ and $|\lambda|$ is the number of elements $\lambda_i$ of the partition.
Is there a word for this recurrence relation? Any key words I could search for related to it?
Edit- I originally thought this sequence would count unlabeled rooted trees of size $n$, but neglected to consider that as written this would count trees with multiple branches of the same size more than once, provided that size was greater than two. The proper recurrence relation for that case would be $$A_{n+1} = \sum_{\lambda(n)}\prod_{k\in\lambda}{A_k-1\choose deg(k)}\frac{1}{A_k-deg(k)}$$ where $deg(k)$ is the number of times $k$ appears in $\lambda$.