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%I #15 Oct 06 2019 09:04:02
%S 1,3,14,81,527,3684,27022,205149,1598303,12705939,102653652,840419676,
%T 6956988612,58132229976,489673597926,4153635860373,35449185841679,
%U 304179698619129,2622657870000646,22710277017073785,197418128701387895
%N Number of Largest Hairpin Family matchings on n edges.
%C The Largest Hairpin Family of matchings is the largest family of matchings formed by repeated edge inflations and vertex insertions into the single edge and the hairpin.
%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.
%F G.f. f satisfies x*f^3 - (2*x+2)*f^2 + 5*f - 3 = 0.
%e a(3) = 14 because of the 15 matchings on 3 edges, only 1 does not lie in the Largest Hairpin Family. In canonical sequence form, the missing matching is given by 121323.
%p f := RootOf(_Z^3*x-2*_Z^2*x-2*_Z^2+5*_Z-3, 1);
%p series(f, x=0, 30);
%K nonn
%O 1,2
%A _Aziza Jefferson_, Mar 25 2015