cp's OEIS Frontend

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

%I A256334 #30 Jun 26 2023 09:29:17
%S A256334 1,1,3,12,51,227,1052,5030,24634,122950,623140,3198502,16593124,
%T A256334 86864578,458294970,2434421685,13008748377,69882215729,377172620330,
%U A256334 2044303447067,11122504636031,60723579401396,332564474286299,1826591420755058,10058928726906713,55528582177881182,307224615377125853,1703330011411361882,9461963582991098963,52655804456941167376,293523046295844013225
%N A256334 Number of C&C Family matchings on n edges.
%C A256334 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.
%H A256334 S. Cao and S.-J. Chen, <a href="http://dx.doi.org/10.1261/rna.1429009">Predicting structures and stabilities for H-type pseudo knots with inter helix loops</a>, RNA 15 (2009), 696-706.
%H A256334 Aziza Jefferson, <a href="http://ufdc.ufl.edu/UFE0047620">The Substitution Decomposition of Matchings and RNA Secondary Structures</a>, PhD Thesis, University of Florida, 2015.
%H A256334 C. Saule, M. Régnier, J.-M. Steyaert, and A. Denise, <a href="http://dx.doi.org/10.1089/cmb.2010.0086">Counting RNA pseudoknotted structures</a>, J. Comput. Biol. 18(10), (2011), 1339-1351.
%F A256334 G.f. f satisfies f = 1 + x*f^2 + (x^2*f^3)/(1-x)^2.
%e A256334 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.
%p A256334 f := RootOf(x^2*_Z^3 + x*(1-x)^2*_Z^2 - (1-x)^2*_Z + (1-x)^2);
%p A256334 series(f, x=0, 30);
%t A256334 f[x_] = Root[x^2 #^3 + x(1-x)^2 #^2 - (1-x)^2 # + (1-x)^2&, 1];
%t A256334 CoefficientList[f[x] + O[x]^31, x] (* _Jean-François Alcover_, Oct 06 2019 *)
%K A256334 nonn
%O A256334 0,3
%A A256334 _Aziza Jefferson_, Mar 25 2015
%E A256334 a(0)=1 prepended by _Alois P. Heinz_, Jul 14 2017