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Search: id:A110235
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| A110235 |
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Triangle read by rows: T(n,k)(1<=k<=n) is the number of peakless Motzkin paths of length n having k (1,0) steps (can be easily translated into RNA secondary structure terminology). |
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+0
2
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| 1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 0, 6, 0, 1, 0, 6, 0, 10, 0, 1, 1, 0, 20, 0, 15, 0, 1, 0, 10, 0, 50, 0, 21, 0, 1, 1, 0, 50, 0, 105, 0, 28, 0, 1, 0, 15, 0, 175, 0, 196, 0, 36, 0, 1, 1, 0, 105, 0, 490, 0, 336, 0, 45, 0, 1, 0, 21, 0, 490, 0, 1176, 0, 540, 0, 55, 0, 1, 1, 0, 196, 0, 1764, 0, 2520, 0
(list; table; graph; listen)
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OFFSET
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1,8
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COMMENT
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Row sums yield A004148. sum(k*T(n,k),k=1..n)=A110236(n).
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REFERENCES
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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
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T(n, k)=[2/(n+k)]binomial((n+k)/2, k)*binomial((n+k)/2, k-1). G.f.=g=g(t, z) satisfies g=1+tzg+z^2*g(g-1).
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EXAMPLE
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T(5,3)=6 because we have UHDHH, UHHDH, UHHHD, HUHDH, HUHHD and HHUHD, where U=(1,1), D=(1,-1), H=(1,0).
Triangle starts:
1;
0,1;
1,0,1;
0,3,0,1;
1,0,6,0,1;
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MAPLE
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T:=proc(n, k) if n+k mod 2 = 0 then 2*binomial((n+k)/2, k)*binomial((n+k)/2, k-1)/(n+k) else 0 fi end: for n from 1 to 14 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A004148, A110236.
Sequence in context: A112743 A119467 A166353 this_sequence A036856 A036855 A147985
Adjacent sequences: A110232 A110233 A110234 this_sequence A110236 A110237 A110238
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 17 2005
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