[seqfan] connected edge-rooted graphs

[Richard J. Mathar]

If A(x) is the the generating function of connected edge-rooted graphs
  with n nodes,
and A126122(x) the generating function of edge-rooted graphs with n nodes
  (not necessarily connected) (is it?),
and A000088(x) the generating function of graphs with n nodes
  (not necessarily connected), is then 
A(x) * A000088(x) = A126122(x) ?

A -> 0, 1, 2, 10, 56, 477, 5879, 117729, 4014125, 242887444, 26562628943... (n>=1)

Can we combine in the same fashion by dividing
A000664 (number of graphs with n edges)
A126133 (number of edge-rooted unlabeled graphs with n edges)
to get B (number of connected edge-rooted unlabeled graphs with n edges) ?
B -> 1, 1, 4, 10, 32, 101, 346, 1220,... (n>=1)
n=1: the connected graph with 1 edge
n=2: the connected linear graph  with 2 edges
n=3: the triangular graph (1 choice),
     the star graph (1 choice)
     the linear graph with 3 edges (2 choices)
n=4: the quadrangle (1 choice)
     the star tree graph (1 choice)
     the linear tree graph with 4 edges (2 choices)
     the triangle with a protruding edge (3 choices)
     the tree of 2-methyl-Butane (3 choices)

RJM