%I A098057
%S A098057 1,1,1,2,4,8,15,27,48,84,147,257,451,796,1413,2526,4544,8226,14978,
%T A098057 27417,50434,93183,172865,321857,601263,1126644,2116968,3987960,7530200,
%U A098057 14249649,27019301,51327965,97676156,186177568,355406479,679425009
%N A098057 Number of peakless Motzkin paths with no U H^j U, no D H^j D and no D 
               H^jU (j>0), where U=(1,1), D=(1,-1) and H=(1,0) (can be easily expressed 
               using RNA secondary structure terminology).
%D A098057 I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary 
               structures, Discrete Appl. Math., 88, 1998, 207-237.
%D A098057 P. R. Stein and M. S. Waterman, On some new sequences generalizing the 
               Catalan and Motzkin numbers, Discrete Math., 26, 1979, 261-272.
%D A098057 M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux et problemes 
               d'enumeration en biologie moleculaire, Publ. I.R.M.A. Strasbourg, 
               1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
%H A098057 M. Vauchassade de Chaumont and G. Viennot, <a href="http://www.mat.univie.ac.at/
               ~slc/opapers/s08viennot.html">Polynomes orthogonaux at problemes 
               d'enumeration en biologie moleculaire</a>, Sem. Loth. Comb. B08l 
               (1984) 79-86.
%F A098057 G.f.=[1-z+z^2-4z^3+2z^4-sqrt(1-2z-z^2+2z^3+z^4-4z^5+4z^6)]/[2z^2*(1-z)^3].
%e A098057 a(4)=4 because we have HHHH, UHDU, HUHD and UHHD; a(6)=15 because from 
               all 17 peakless Motzkin paths of length 6 (see A004148) only (UHU)HDD 
               and UUH(DHD) do not qualify.
%Y A098057 Cf. A004148.
%Y A098057 Sequence in context: A001523 A000126 A143281 this_sequence A074029 A138653 
               A054159
%Y A098057 Adjacent sequences: A098054 A098055 A098056 this_sequence A098058 A098059 
               A098060
%K A098057 nonn
%O A098057 0,4
%A A098057 Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 11 2004