TITLE: On the number of ways to draw kissing circles
QUESTION [29 upvotes]: So I was watching Numberphile with Neil Sloane of OEIS fame on the number of ways to make circles intersect. During the intro, he explicitly forbid kissing (touching, tangent) circles. This made me think about the number of ways to draw circles that must all kiss, where intersections are not allowed. Kind of the dual problem, if you will. So in a nutshell:

Circles in the affine plane can be of any finite size
Circles must kiss at least one other (no separate circles, all must be connected)
Solutions that can be transformed into each other through mirroring, rotation, stretching, moving circles without making new tangent points, are considered the same

For $n=2$ circles there are two ways: touching on the outside, and touching on the inside of the larger circle.
For $n=3$ I believe the answer is 9. Try it, should be easy.
For $n=4$ I started drawing and came up with, so far, and with help from you guys, 38, 39, 42, 43, 44, 45, 47, 44 ways. The colored dots represent the number of kissing circles at the tangent points.
Questions:

Did I miss any? I'm sure I did.
Do I have inadvertent duplicates?
Does an integer sequence exist, or even a formula mapping $n = 2, 3, 4, ...$ to $2, 9, 44, ...$?

REPLY [10 votes]: Thanks to @DanUznanski here is the complete solution with 58 ways, using his sibling relation to group diagrams.