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A098076
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Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n and height k (can be easily expressed using RNA secondary structure terminology).
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0
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1, 1, 1, 1, 1, 1, 3, 1, 6, 1, 1, 11, 5, 1, 20, 15, 1, 1, 36, 38, 7, 1, 64, 91, 28, 1, 1, 113, 211, 89, 9, 1, 199, 477, 255, 45, 1, 1, 350, 1059, 690, 172, 11, 1, 615, 2321, 1797, 572, 66, 1, 1, 1080, 5037, 4555, 1754, 295, 13, 1, 1896, 10847, 11320, 5098, 1118, 91, 1, 1, 3328
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,7
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COMMENTS
| Row sums are the RNA secondary structure numbers (A004148). Column 1 without the zeros yields A077855.
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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.
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LINKS
| M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux at problemes d'enumeration en biologie moleculaire, Sem. Loth. Comb. B08l (1984) 79-86.
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FORMULA
| G.f. = 1+sum(t^j*z^(2j+1)/[P(j)*P(j+1)], j=0..infinity), where P(j) are polynomials in z defined by P(0)=1, P(1)=1-z, P(j)=(1-z+z^2)P(j-1)-z^2*P(j-2), j=2, 3, ....
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EXAMPLE
| Triangle starts:
1;
1;
1;
1,1;
1,3;
1,6,1;
1,11,5;
1,20,15,1;
Row n >0 has ceil(n/2) terms.
T(6,2)=5 because the peakless Motzkin paths of length 6 and height 2 are HUUHDD, UHUHDD, UUHHDD, UUHDDH, UUHDHD, where U=(1,1), H=(1,0) and D=(1,-1).
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MAPLE
| P[0]:=1: P[1]:=sort(1-z): for j from 2 to 30 do P[j]:=sort(expand((1-z+z^2)*P[j-1]-z^2*P[j-2])) od: G:=1+sum(t^i*z^(2*i+1)/P[i]/P[i+1], i=0..25): Gser:=simplify(series(G, z=0, 21)): Q[0]:=1: for m from 1 to 18 do Q[m]:=sort(coeff(Gser, z^m)) od: 1, seq(seq(coeff(t*Q[n], t^k), k=1..ceil(n/2)), n=1..16);
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CROSSREFS
| Cf. A004148, A077855.
Sequence in context: A034839 A089732 A158905 * A171852 A010287 A130270
Adjacent sequences: A098073 A098074 A098075 * A098077 A098078 A098079
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 13 2004
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