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%I #22 Oct 06 2019 09:04:53
%S 1,3,14,81,526,3655,26522,198322,1516296,11794717,93028387,742192059,
%T 5978650560,48558821234,397218622275,3269629207524,27061726430000,
%U 225078993453143,1880240716499975,15768890757767329,132719696885282352,1120664726059889642,9490737694928103944,80593740187789336604,686097231181385302494,5854230604182513256777,50058728487687099021228,428893610758038945556024,3681458291424994103104272,31654643493605098603330050,272617697673293256259943417,2351397730980411031399548438,20310185543805378949877753778,175663385844074502933143530174,1521230708939544454165789841800,13189400713003422051741601456307,114483609078595784724427186310842,994773380472692869438699360298740,8652545469871591210786412806190538
%N Number of A&U Family matchings on n edges.
%C The A&U Family of matchings is the largest family of matchings formed by pinning edges to the right, edge inflation by ladders and vertex insertions.
%H T. Akutsu, <a href="http://dx.doi.org/10.1016/S0166-218X(00)00186-4">Dynamic programming algorithms for RNA secondary structure prediction with pseudoknots</a>, Discrete Appl. Math. 104(1-2), (2000), 45-62.
%H A. Condon, B. Davy, B. Rastegari, S. Zhao and F. Tarrant, <a href="http://dx.doi.org/10.1016/j.tcs.2004.03.042">RNA pseudoknotted structures</a>, Theoret. Comput. Sci. 320(1), (2004), 35-50.
%H 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 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.
%H Y. Uemura, A. Hasegawa, S. Kobayashi, and T. Yokomori, <a href="http://dx.doi.org/10.1016/S0304-3975(98)00090-5">Tree adjoining grammars for RNA structure prediction</a>, Theoret. Comput. Sci. 210(2), (1999), 277-303.
%F G.f. f satisfies 2x^7f^14 - 2x^6f^13 - 3x^6f^12 + 7x^5f^11 + 3x^5f^10 - 16x^4f^9 + 2x^4f^8 + 18x^3f^7 - 7x^2f^5 + (-12x^3 + 2x^2)f^6 + (4x^2 - 5x)f^4 + 10xf^3 + (-5x+1)f^2 - 2f + 1 = 0.
%e a(4)=81 because of the 105 matchings on 4 edges which can be drawn in the plane, 24 do not lie in the A&U Family. Of these 24, only three lie in the R&E family. In canonical sequence form the three missing matchings are given by 12134324, 12324314, and 12343142.
%p f := RootOf(2*x^7*_Z^(14)-2*x^6*_Z^(13)-3*x^6*_Z^(12)+7*x^5*_Z^(11)+3*x^5*_Z^(10)-16*x^4*_Z^9+2*x^4*_Z^8+18*x^3*_Z^7-7*x^2*_Z^5+(-12*x^3+2*x^2)*_Z^6+(4*x^2-5*x)*_Z^4+10*x*_Z^3+(-5*x+1)*_Z^2-2*_Z+1); convert(series(f, x=0, 40), radical);
%Y Cf. A256330, A256337.
%K nonn
%O 1,2
%A _Aziza Jefferson_, Mar 29 2015
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