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A089737 Number of (1,1) steps starting at level zero in all peakless Motzkin paths of length n+3 (can be easily expressed also in RNA secondary structure terminology). +0
2
1, 3, 7, 17, 41, 98, 235, 565, 1362, 3294, 7992, 19450, 47475, 116204, 285178, 701585, 1730003, 4275162, 10586164, 26263365, 65273566, 162499838, 405185762, 1011815774, 2530219435, 6335642377, 15884284791, 39871297479, 100194076029 (list; graph; listen)
OFFSET

0,2

COMMENT

lim(a(n)/A004148(n), n=infinity) = sqrt(5).

REFERENCES

I. L. Hofacker, P. Schuster, and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.

P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26, 1979, 261-272.

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.

LINKS

M. S. Waterman, Home Page (contains copies of his papers)

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

FORMULA

a(n)=sum((5k-2n-2)binomial(k, n+1-k)*binomial(k+1, n+3-k)/[k(n+4-k)], k=ceil(n/2+1)..n+1). a(n)=A004148(n+5)-2A004148(n+4)+A004148(n+3)-A004148(n+2). G.f.=2/[1-3z+2z^2-2z^3+2z^4-z^5+(1-2z+z^2-z^3)sqrt(1-2z-z^2-2z^3+z^4)].

EXAMPLE

a(2)=7 because in the eight peakless Motzkin paths of length 5, namely HHHHH, HHU'HD, HU'HHD, HU'HDH, U'HDHH, U'HHDH, U'HHHD, and U'UHDD, where U=(1,1), D=(1,-1), H=(1,0), we have alltogether seven U steps starting at level zero (indicated by U').

CROSSREFS

Cf. A004148.

Adjacent sequences: A089734 A089735 A089736 this_sequence A089738 A089739 A089740

Sequence in context: A000600 A131056 A077851 this_sequence A123335 A001333 A078057

KEYWORD

nonn

AUTHOR

Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 07 2004

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Last modified October 7 14:39 EDT 2008. Contains 144666 sequences.


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