@johncarlosbaez
oh i just accidentally ran into these the number of rooted 2 connected planar graphs with n edges (allowing multiple edges between vertices) runs as G_2=1,2,6,22,91 whereas the schroder numbers run 1,2,6,22,91.
roughly you can build 2-connected planar graphs out of 3-connected planar graphs and the schroder numbers count graphs which are "trivial" in this sense
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@johncarlosbaez
"build": glue 2 graphs along an edge and remove this edge
"trivial": anything you can build out of k-cycles and the graph with 2 vertices joined by k edges (k>=2)
@johncarlosbaez
(i'd had in my head the problem of counting 3-connected planar graphs for a while, just learning someone else had done something like it helped me put the pieces together)