A256334 Number of C&C Family matchings on n edges.

1, 1, 3, 12, 51, 227, 1052, 5030, 24634, 122950, 623140, 3198502, 16593124, 86864578, 458294970, 2434421685, 13008748377, 69882215729, 377172620330, 2044303447067, 11122504636031, 60723579401396, 332564474286299, 1826591420755058, 10058928726906713, 55528582177881182, 307224615377125853, 1703330011411361882, 9461963582991098963, 52655804456941167376, 293523046295844013225

Offset: 0

Author

Aziza Jefferson, Mar 25 2015

Keywords

• nonn

Comments

The C&C Family of matchings is the family of matchings formed by first vertex insertions into the hairpin (except beneath both edges) or single edge (as long as the inserted edge does not have an outer edge connecting the first and last vertex), then edge inflations by ladders of the original single edge or hairpin.

Examples

			a(3)=12 because of the 15 matchings on 3 edges, three do not lie in the C&C Family. In canonical sequence form the missing matchings are given by 121323, 123123, and 123312.
		

Links

• S. Cao and S.-J. Chen, Predicting structures and stabilities for H-type pseudo knots with inter helix loops, RNA 15 (2009), 696-706.
• Aziza Jefferson, The Substitution Decomposition of Matchings and RNA Secondary Structures, PhD Thesis, University of Florida, 2015.
• C. Saule, M. Régnier, J.-M. Steyaert, and A. Denise, Counting RNA pseudoknotted structures, J. Comput. Biol. 18(10), (2011), 1339-1351.

Programs

Maple

f := RootOf(x^2*_Z^3 + x*(1-x)^2*_Z^2 - (1-x)^2*_Z + (1-x)^2);
series(f, x=0, 30);

Mathematica

f[x_] = Root[x^2 #^3 + x(1-x)^2 #^2 - (1-x)^2 # + (1-x)^2&, 1];
CoefficientList[f[x] + O[x]^31, x] (* Jean-François Alcover, Oct 06 2019 *)

Formula

G.f. f satisfies f = 1 + x*f^2 + (x^2*f^3)/(1-x)^2.

Extensions

a(0)=1 prepended by Alois P. Heinz, Jul 14 2017