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A097777
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Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k U H^j Us for some j>0, where U=(1,1) and H=(1,0) (can be easily expressed using RNA secondary structure terminology).
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3
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1, 1, 1, 2, 4, 8, 16, 1, 32, 5, 65, 17, 134, 50, 1, 280, 136, 7, 592, 355, 31, 1264, 904, 114, 1, 2722, 2264, 378, 9, 5906, 5604, 1176, 49, 12900, 13752, 3504, 215, 1, 28344, 33530, 10112, 835, 11, 62608, 81358, 28468, 2997, 71, 138949, 196688, 78576, 10173, 361, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Row n contains floor(n/3) entries (n>=3).
Row sums are the RNA secondary structure numbers (A004148).
T(n,0)=A098051(n).
Sum(k*T(n,k), k>=0)=A187257(n).
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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. = G = G(t, z) satisfies G=1+zG+z^2*G[G-1-(1-t)[zG-z/(1-z)]].
The generating function H=H(t,z) relative to the number of subwords of the form UH^bU for a fixed b>=1 satisfies H = 1+zH+z^2*H[H-1+(t-1)z^b*(H-1-zH)].
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EXAMPLE
| Triangle starts:
1;
1;
1;
2;
4;
8;
16,1;
32,5;
65,17;
134,50,1;
280,136,7;
Row n has floor(n/3) terms, n>=3.
T(7,1)=5 because we have H(UHU)HDD, (UHU)HHDD, (UHU)HDHD, (UHU)HDDH and (UHHU)HDD, where U=(1,1), H=(1,0) and D=(1,-1); the U H^j U's are shown between parentheses.
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MAPLE
| eq := G = 1+z*G+z^2*G*(G-1+(t-1)*(z*G-z/(1-z))): g := RootOf(eq, G): gser := simplify(series(g, z = 0, 23)): for n from 0 to 18 do P[n] := sort(coeff(gser, z, n)) end do: 1; 1; 1; for n from 3 to 18 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)-1) end do; # yields sequence in triangular form
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CROSSREFS
| Cf. A004148, A098051, A187257
Sequence in context: A098864 A002546 A010745 * A089738 A110333 A069783
Adjacent sequences: A097774 A097775 A097776 * A097778 A097779 A097780
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 11 2004
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