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A110237
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Triangle read by rows: T(n,k) (0<=k<=ceil(n/2)-1) is the number of (1,0) steps at level k in all peakless Motzkin paths of length n (can be easily translated into RNA secondary structure terminology).
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1
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1, 2, 3, 1, 6, 4, 13, 10, 1, 28, 24, 6, 62, 59, 21, 1, 140, 144, 62, 8, 320, 350, 174, 36, 1, 740, 852, 474, 128, 10, 1728, 2077, 1263, 410, 55, 1, 4068, 5072, 3318, 1240, 230, 12, 9645, 12412, 8634, 3608, 835, 78, 1, 23010, 30440, 22314, 10216, 2792, 376, 14
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Row n has ceil(n/2) terms. Row sums yield A110236.
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REFERENCES
| W. R. Schmitt and M. S. Waterman, Linear trees and RNA secondary structure, Discrete Appl. Math., 51, 317-323, 1994.
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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FORMULA
| G.f.=z*g^2/(1-tz^2*g^2), where g=1+zg+z^2*g(g-1)=[1-z+z^2-sqrt(1-2z-z^2-2z^3+z^4)]/(2z^2) is the g.f. of the RNA secondary structure numbers (A004148).
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EXAMPLE
| T(5,1)=10 because in the 8 (=A004148(5)) peakless Motzkin paths of length 5, namely HHHHH, U(H)DHH, U(HH)DH, U(HHH)D, HU(H)DH, HU(HH)D, HHU(H)D and UUHDD (where U=(1,1), H=(1,0) and D=(1,-1)), we have alltogether 10 H steps at level 1 (shown between parentheses).
Triangle starts:
1;
2;
3,1;
6,4;
13,10,1;
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MAPLE
| g:=(1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))/2/z^2: G:=z*g^2/(1-t*z^2*g^2): Gser:=simplify(series(G, z=0, 20)): for n from 1 to 15 do P[n]:=coeff(Gser, z^n) od: for n from 1 to 15 do seq(coeff(t*P[n], t^k), k=1..ceil(n/2)) od;
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CROSSREFS
| Cf. A004148, A110236.
Sequence in context: A205112 A173161 A116468 * A189970 A076631 A035485
Adjacent sequences: A110234 A110235 A110236 * A110238 A110239 A110240
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 17 2005
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