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%I #4 Mar 30 2012 17:35:59
%S 1,1,1,1,1,1,2,1,1,4,2,1,1,6,7,2,1,1,9,13,11,2,1,1,12,28,22,16,2,1,1,
%T 16,46,64,33,22,2,1,1,20,80,118,126,46,29,2,1,1,25,120,258,248,225,61,
%U 37,2,1,1,30,185,438,668,460,374,78,46,2,1,1,36,260,813,1231,1506,782,588,97
%N Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n and having k level steps at height >0 (can be easily expressed using RNA secondary structure terminology).
%C Row sums yield the RNA secondary structure numbers (A004148).
%D I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
%D P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26, 1979, 261-272.
%D 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 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 G.f.=G=G(t, z) satisfies z(t-tz+tz^2-1+2z-z^2)G^2-(1-2z+z^2+tz)G+1=0.
%e Triangle starts:
%e 1;
%e 1;
%e 1;
%e 1,1;
%e 1,2,1;
%e 1,4,2,1;
%e 1,6,7,2,1;
%e 1,9,13,11,2,1;
%e Row n (n>=2) has n-1 terms.
%e T(5,2)=2 because among the eight peakless Motzkin paths of length 5 only HU(HH)D and U(HH)DH have two H's at positive height (shown between parentheses); here U=(1,1), H=(1,0), D=(1,-1).
%Y Cf. A004148.
%K nonn,tabf
%O 0,7
%A _Emeric Deutsch_, Sep 12 2004
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