Arranging Bubbles

As Arthur points out in a comment, this is A000081 in the OEIS, "Number of unlabeled rooted trees with $n$ nodes . . . Also, number of ways of arranging $n-1$ nonoverlapping circles".

Selected formulas from the linked page:

Generating function $A(x) = \sum\limits_{n\ge1} a(n) x^n$ satisfies $$A(x) = x\exp\left(A(x)+\frac12A(x^2)+\frac13A(x^3)+\frac14A(x^4)+\cdots\right)$$ [Polya]

Also $$A(x) = \frac x{\prod\limits_{n\ge1} (1-x^n)^{a(n)}}.$$

Recurrence: $$a(n+1) = \frac1n \sum_{k=1}^n \Bigl( \sum_{d|k} da(d) \Bigr) a(n-k+1).$$

Asymptotically $c d^n n^{-3/2}$, where $c = 0.439924\ldots$ and $d = 2.955765\ldots$ [Polya; Knuth, section 7.2.1.6].

Reminder: $a(n)$ is the number of ways of arranging $n-1$ circles, not $n$ circles. So we have $a(5)=9$.