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

0,3

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.

Links

Table of n, a(n) for n=0..30.

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.

Formula

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

Example

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.

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 *)

Crossrefs

Sequence in context: A110167 A151317 A151184 * A151185 A151186 A151187

Adjacent sequences: A256331 A256332 A256333 * A256335 A256336 A256337

Keyword

nonn

Author

Aziza Jefferson, Mar 25 2015

Extensions

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

Status

approved