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

[franktaw at netscape.net]

There are actually two different transforms one might apply, depending 
on whether we want to consider the order of the piles as significant.  
If it isn't, this is the Euler transform.  For this I get (from 0):

1,1,2,4,9,18,37,75

I don't remember what the transform is called for the case when the 
order is significant, but the result is:

1,1,2,5,13,33,82,207

Franklin T. Adams-Watters

-----Original Message-----
From: Jonathan Post <jvospost3 at gmail.com>

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