[seqfan] Possibly disconnected piles whose subpiles are as in A168368 Number of stable connected piles of n bricks.

[Jonathan Post]

Doodling on paper, I suspect that the other case suggested (where
subpiles have nonnegative number of bricks) is:

"Connected piles only (allowing piles with disconnected subpiles would
produce a different sequence.)"

1, 2, 4, 8, 15, 26, 47

I get this as follows:

a(1) = 1
a(2) = 2 because one may have the "2" pile, or "1" and another subpile
"1"  -- I'll denote this {"2", "1" + "1"}
a(3) = 4 = card{"3_1", "3_2), any of the a(2) solutions + "1}  where
3_1 is the first solution with 3 bricks and 3_2 is the 2nd solution
a(4) = 8 = card{"4_1", "4_2", "4_3", "4_4", any of the a(s) solutions + "1"

wait, I'm wrong.  I can get 4 bricks as "2" + "2" as well.  Okay,
different transform.

We want a(n) = the number of non-isomorphic partitions of n into
non-zero values in A168368.

I forget the name of this partition-like transform.  I'd redo this,
but need to share dinner with my wife now.

-- Prof. Jonathan Vos Post