Matryoshka Configurations

Ways to arrange N nesting dolls when each doll comes apart into a top and a bottom half. Every configuration for N = 1 to 4 is drawn below.

The rules

Doll k is either closed (top on bottom) or split into its two halves.
A piece only goes inside a larger doll's piece.
A closed doll or an open bottom holds one item directly: a smaller closed doll or a smaller bottom (which may hold more). Nothing sits side by side inside a doll.
An open top holds one item directly, and only a smaller top. A top never covers a doll or a bottom.
A loose top never goes inside a bottom or a closed doll, even if its own bottom is also there.
Position on the table doesn't matter. Only what is inside what counts.

The sequence

Further terms, N = 1…16:

2, 10, 75, 780, 10556, 178031, 3630780, 87548580, 2452523325, 78697155750, 2859220516290, 116482516809889, 5277304280371714, 264005848594606490, 14493602135008296115, 868435614538568029188

This is A124426, the product of two consecutive Bell numbers: a(N) = B(N)·B(N+1). The bottoms and closed dolls form stacks, and any grouping of the N dolls into stacks works. That gives B(N) ways. Each stack's order is forced by size. The loose tops then stack among themselves. Choosing which dolls are open and how their tops stack gives B(N+1) ways in total. A brute-force enumeration agrees for N ≤ 6.