[seqfan] Help wanted on A005646 connection to A000055

[Robert Munafo]

Andrew Weimholt and I have been working on A005646. Franklin T.
Adams-Watters suggested the importance of the triangle:

  N=1   1
  N=2   0  1
  N=3   0  0  1
  N=4   0  0  1  2
  N=5   0  0  0  3   3
  N=6   0  0  0  3  17    6
  N=7   0  0  0  1  36   74   11
  N=8   0  0  0  1  60  573  358   23
  N=9   0  0  0  0  56 2802 7311 1631  47

The main diagonal suggests a relation to sequence A000055, unlabaled
unrooted trees.

I am starting to convince myself such a connection is real; detals and full
examples through N=6 are at http://mrob.com/pub/math/seq-a005646.html#trees

A graph edge corresponds to a binary partition, and each node is one of the
types in the classification. Each edge connects two subgraphs, and the
corresponding partition divides the nodes in the same way the graph gets
divided if that edge is removed.

Can anyone help me prove or disprove this?

-- 
 Robert Munafo  --  mrob.com