0,5

Number of Motzkin paths of length n-1 with no peaks, no double rises, and no doubledescents (i.e. no UD's, no UU's, and no DD's, where U=(1,1) and D=(1,-1), n>0 (can be easily formulated using RNA secondary structure terminology). E.g. a(5)=4 because we have HHHH, HUHD, UHDH, and UHHD; here H=(1,0). Also number of peakless Motzkin paths of length n in which each D=(1,-1) step is followed by a H=(1,0) step (can be easily formulated using RNA secondary structure terminology). E.g. a(5)=4 because we have HHHHH, HUHDH, UHDHH, and UHHDH (here U=(1,1)). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 09 2004

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.

M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux at problemes d'enumeration en biologie moleculaire, Sem. Loth. Comb. B08l (1984) 79-86.

G.f.=[1-z+z^3-sqrt(1-2z-2z^3+z^2-2z^4+z^6)]/(2z^3). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 09 2004

Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-3-k ], {k, 0, n-4} ];

Cf. A000108, A001006, A004148, A006318.

Sequence in context: A000072 A018179 A127542 this_sequence A072641 A135360 A082548

Adjacent sequences: A023429 A023430 A023431 this_sequence A023433 A023434 A023435

nonn,easy

Olivier Gerard (ogerard(AT)ext.jussieu.fr)