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A089741
Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k UHH...HD's, where U=(1,1), D=(1,-1) and H=(1,0) (can be easily expressed using RNA secondary structure terminology).
1
1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 15, 1, 1, 31, 5, 1, 63, 18, 1, 127, 56, 1, 1, 255, 160, 7, 1, 511, 432, 34, 1, 1023, 1120, 138, 1, 1, 2047, 2816, 500, 9, 1, 4095, 6912, 1672, 55, 1, 8191, 16640, 5264, 275, 1, 1, 16383, 39424, 15808, 1205, 11, 1, 32767, 92160, 45696, 4797
OFFSET
0,7
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
G.f.=[1-2z+2z^2-tz^3-sqrt[(1-tz^3)*(1-4z+4z^2-tz^3)]]/[2z^2*(1-z].
EXAMPLE
T(7,2)=5 because we have H(UHD)(UHD), (UHD)H(UHD), (UHD)(UHD)H, (UHD)(UHHD) and (UHHD)(UHD) (the required subwords are shown between parentheses).
1; 1; 1; 1,1; 1,3; 1,7; 1,15,1; 1,31,5; 1,63,18; 1,127,56,1; 1,255,160,7;
CROSSREFS
Row sums yield A004148. Column 1 is A000225, column 2 is A001793.
Sequence in context: A160627 A114712 A321452 * A089736 A205479 A094024
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jan 08 2004
STATUS
approved