Sequence A003238 of the OEIS counts ``rooted trees with $n$ vertices in which vertices at the same level have the same degree.'' The sequence, $a$, begins

1, 1, 2, 3, 5, 6, 10, 11, 16, ...

and it is of course easy to generate as many terms as you like via recurrences (basically $a(n)$ is the sum of $a(d)$ as $d$ runs over divisors of $n-1$). In the formula section of the OEIS page we find:

Conjecture : $\log(a(n))$ is asymptotic to $c \log(n)^2$ where $0.4 < c < 0.5$ (Benoit Cloitre, Apr 13 2004)

What I'm interested in is both the state of this conjecture, and more generally, methods for analysing the asymptotics of sequences defined by ``similar'' recurrences - either globally (as above) or in the average sense i.e. asymptotics of things like $(1/n)\sum_{i=1}^n a(n)$.