Not quite an answer, but two possible sources:

(1) A003094 lists the number of unlabeled connected planar simple graphs with $n$ nodes.

(2) Read, Ronald C., and Robin J. Wilson. An atlas of graphs. Clarendon Press, 1998. I can't access more than the table of contents of this book at the moment:

Not quite an answer, but two possible sources:

(1) A003094 lists the number of unlabeled connected planar simple graphs with $n$ nodes: $$1, 1, 1, 2, 6, 20, 99, 646, 5974, 71885, 1052805, 17449299, 313372298\;.$$

(2) Read, Ronald C., and Robin J. Wilson. An atlas of graphs. Clarendon Press, 1998. I can't access more than the table of contents of this book at the moment: