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 A256333 Number of R&G Family matchings on n edges. 0
 1, 3, 13, 61, 301, 1552, 8277, 45284, 252753, 1433633, 8239993, 47887467, 280927846, 1661387046, 9894376821, 59288650788, 357198545904, 2162437157263, 13147835385477, 80251977589719, 491573099486143, 3020738578507674, 18617035563669489, 115046892012376542, 712710925868858139, 4425312432316379040, 27535525144298975942, 171670784266383750322, 1072246008621559982926, 6708644077265798380125 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The R&G Family of matchings is the family of matchings formed by first vertex insertions into the hairpin 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 Aziza Jefferson, The Substitution Decomposition of Matchings and RNA Secondary Structures, PhD Thesis, University of Florida, 2015. J. Reeder and R. Giegerich, Design, implementation and evaluation of a practical pseudo knot folding algorithm based on thermodynamics, BMC Bioinform. 5 (2004), Article #104. 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 x^2f^4 + x(1-x)^2f^2 - (1-x)^2f + (1-x)^2. EXAMPLE a(3)=13 because of the 15 matchings on 3 edges, two do not lie in the R&G Family. In canonical sequence form the missing matchings are given by 121323 and 123123. a(4)= 61 out of the 105 matchings on 4 edges, one such matching which does not lie in the R&G Family is given by 12234314. MAPLE f := RootOf(x^2*_Z^4 + x*(1-x)*(_Z-x*_Z)*_Z - (1-x)^2*_Z + (1-x)^2); series(f, x=0, 30); CROSSREFS Sequence in context: A341250 A026704 A046748 * A200215 A074548 A243014 Adjacent sequences:  A256330 A256331 A256332 * A256334 A256335 A256336 KEYWORD nonn AUTHOR Aziza Jefferson, Mar 25 2015 STATUS approved

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Last modified August 11 07:13 EDT 2022. Contains 356046 sequences. (Running on oeis4.)