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A098086
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Triangle read by rows: T(n,k) is the number of peakless Motzkin paths in which the k-th step is the leftmost (1,0)-step (can be easily expressed using RNA secondary structure terminology).
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0
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1, 1, 1, 1, 2, 2, 4, 3, 1, 8, 6, 3, 17, 13, 6, 1, 37, 28, 13, 4, 82, 62, 30, 10, 1, 185, 140, 69, 24, 5, 423, 320, 160, 59, 15, 1, 978, 740, 375, 144, 40, 6, 2283, 1728, 885, 350, 105, 21, 1, 5373, 4068, 2102, 852, 271, 62, 7, 12735, 9645, 5022, 2077, 690, 174, 28, 1, 30372
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Row sums yield the RNA secondary structure numbers (A004148).
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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
| T(n, k)=k*sum((1/j)binomial(j, n-2k+1-j)*binomial(j+k-1, n-k+1-j), j=ceil((n-2k+2)/2)..n-2k+1) if 2k<n+1 and T(n, k)=1 if 2k=n+1. G.f.=tzg/(1-tz^2*g), where g = [1-z+z^2-sqrt(1-2z-z^2-2z^3+z^4)]/(2z^2) is the g.f. for the RNA secondary structure numbers (A004148).
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EXAMPLE
| Triangle starts:
1;
1;
1,1;
2,2;
4,3,1;
8,6,3;
17,13,6,1;
Row n has ceil(n/2) terms.
T(5,2)=3 because we have UHDHH, UHHDH and UHHHD, where U=(1,1), H=(1,0) and
D=(1,-1); these are the only peakless Motzkin paths of length 5 in which the second step is the first occurrence of H.
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MAPLE
| T:=proc(n, k) if 2*k-1=n then 1 else k*sum(binomial(j, n-2*k+1-j)*binomial(j+k-1, n-k+1-j)/j, j=ceil((n-2*k+2)/2)..n-2*k+1) fi end: seq(seq(T(n, k), k=1..ceil(n/2)), n=0..16); # yields the sequence in linear form T:=proc(n, k) if 2*k-1=n then 1 else k*sum(binomial(j, n-2*k+1-j)*binomial(j+k-1, n-k+1-j)/j, j=ceil((n-2*k+2)/2)..n-2*k+1) fi end: matrix(10, 10, T); # yields the sequence in triangular form
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CROSSREFS
| Cf. A004148.
Sequence in context: A011388 A105114 A166284 * A175681 A161003 A152028
Adjacent sequences: A098083 A098084 A098085 * A098087 A098088 A098089
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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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