The number of nearly 4-regular planar maps (Q2707967)

In [Enumeration of quadrangular dissections of the disk, Can. J. Math. 17, 302-317 (1965; Zbl 0138.19104)] the reviewer investigated the numbers of rooted dissections of the disk into quadrangles, and obtained a closed form formula for these numbers, generalizing the 1791 result of \textit{N. Fuss} [Solutio quæstionis quot modis polygonum \(n\) laterum in polygona \(m\) laterum per diagonales resolvi queat, Nova Acta Acad. Sci. Imp. Petropolitanæ 9, 243-251 (1791)] in which there were no internal vertices. In the present paper the authors solve several problems dually related to this problem: the rooted maps have all but at most one vertex of degree 4. Their method derives from the parametric solution of certain functional equations for generating functions; all of the sequences determined require summations in their description.