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
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
KEYWORD
nonn
AUTHOR
Aziza Jefferson, Mar 25 2015
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Jul 14 2017
STATUS
approved