%I
%S 1,4,15,80,362,1832,8994,46384,238838,1257824
%N Number of rooted generalized quadrangular dissections of weight n of a closed disk: planar maps having the external face bounded by a polygon and all internal faces of size 4.
%C Some boundary mutually nonadjacent nodes of valency 2 are marked as singular; (boundary) edges incident to them are also called singular. The maps are considered up to rotations and reflections. Rooting means distinguishing a nonsingular edge, an end and an internal side of it. n is the number of internal edges plus half of the number of nonsingular boundary edges.
%C No formula is known. For any generalized quadrangular dissection, s==n (mod 2), where s is the number of singular nodes.
%H V. A. Liskovets, <a href="http://dx.doi.org/10.1016/0012365X(94)00347L">A reductive technique for enumerating nonisomorphic planar maps</a>, Discr. Math., v.156 (1996), 197217.
%e The four rooted generalized quadrangular dissections of weight 1 are
%e ...................____......____..
%e .X<X..X<X.../....\..../....\.
%e .............X<X..O..X<X..O
%e ..............\____/....\____/.
%e .XO..XO....................
%e where O is the singular node and > is the rooted edgeend.
%Y Cf. A006385.
%K nonn,more
%O 0,2
%A _Valery A. Liskovets_, Dec 03 2003
