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Search: id:A110317
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| A110317 |
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Triangle read by rows: T(n,k) (k>=0) is the number of RNA secondary structures of size n (i.e. with n nodes) having k arcs that are covered by other arcs. |
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+0
2
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| 1, 1, 1, 2, 4, 7, 1, 12, 5, 21, 15, 1, 37, 37, 8, 65, 84, 35, 1, 114, 182, 115, 12, 200, 381, 323, 73, 1, 351, 777, 825, 313, 17, 616, 1554, 1977, 1087, 138, 1, 1081, 3062, 4524, 3291, 754, 23, 1897, 5962, 9999, 9063, 3209, 241, 1, 3329, 11496, 21515, 23300
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Rows 0,1,2 have one term each; row n>=3 has ceil(n/2)-1 terms. Rows sums yield A004148. T(n,0)=A005251(n+1). Sum(k*T(n,k),k>=0)=A110318(n-5)
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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, 1978, 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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G.f.=2t/[2t-2tz-1+z+tz^2+sqrt(1-2z-2tz^2+z^2-2tz^3+t^2*z^4)].
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EXAMPLE
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T(6,1)=5 because we have 15/(24)/3/6, 16/(24)/3/5, 16/(25)/3/4, 16/2/(35)/4, and 1/26/(35)/4 (the covered arcs are shown between parentheses).
Triangle begins
1;
1;
1;
2;
4;
7,1;
12,5;
21,15,1;
37,37,8;
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MAPLE
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G:=2*t/(2*t-2*z*t-1+z+t*z^2+sqrt(1-2*z-2*t*z^2+z^2-2*t*z^3+t^2*z^4)): Gser:=simplify(series(G, z=0, 20)): P[0]:=1: for n from 1 to 17 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 2 do print(1) od: for n from 3 to 17 do seq(coeff(t*P[n], t^k), k=1..ceil(n/2)-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A004148, A005252, A110318.
Sequence in context: A089349 A118424 A118429 this_sequence A098073 A118390 A134974
Adjacent sequences: A110314 A110315 A110316 this_sequence A110318 A110319 A110320
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 19 2005
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