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Search: id:A110320
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| A110320 |
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Number of blocks in all RNA secondary structures with n nodes (an RNA secondary structure can be viewed as a restricted noncrossing partition). |
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+0
4
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| 1, 2, 5, 13, 32, 80, 201, 505, 1273, 3217, 8146, 20668, 52531, 133726, 340909, 870213, 2223958, 5689807, 14571335, 37350585, 95821071, 246015677, 632088930, 1625119218, 4180845277, 10762096850, 27718352411, 71426753423, 184146711578
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(n)=Sum(k*A110319(n,k), k=1..n).
Conjecture: A110320(n) = (A051292(n+2)-A051286(n+1))/2. - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Jan 14 2007
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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.=(1-z-z^2)/[2z^2*sqrt(1-2z-z^2-2z^3+z^4)]-1/(2z^2).
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EXAMPLE
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a(4)=13 because the 4 (=A004148(4)) RNA secondary structures of size 4, namely 1/2/3/4, 13/2/4, 14/2/3, and 1/24/3, have alltogether 4+3+3+3=13 blocks.
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MAPLE
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G:=1/2*(1-z-z^2)/z^2/(1-2*z-z^2-2*z^3+z^4)^(1/2)-1/2*1/(z^2): Gser:=series(G, z=0, 37): seq(coeff(Gser, z^n), n=1..33);
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CROSSREFS
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Cf. A004148, A110319.
Sequence in context: A116702 A098156 A098586 this_sequence A108890 A027929 A001659
Adjacent sequences: A110317 A110318 A110319 this_sequence A110321 A110322 A110323
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 19 2005
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