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A202842
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Number of secondary structures of size n having no stacks of length 2.
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6
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1, 1, 1, 2, 4, 7, 14, 31, 66, 142, 316, 708, 1593, 3625, 8314, 19165, 44433, 103557, 242376, 569514, 1343099, 3177766, 7540845, 17943506, 42804078, 102345017, 245233366, 588785677, 1416247791, 3412495415, 8235829927, 19906780104, 48185131721, 116790380824, 283432579807
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OFFSET
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0,4
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COMMENTS
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For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
a(n) = A202841(n,0).
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REFERENCES
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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.
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LINKS
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Table of n, a(n) for n=0..34.
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FORMULA
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G.f. G=G(z) satisfies G = 1+zG +fG(G-1)/(1+f), where f = z^2*(1-z^2+z^4)/(1-z^2).
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EXAMPLE
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a(5)=7; representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv, ABvBA; only the last one has a stack of length 2.
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MAPLE
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f := z^2*(1-z^2+z^4)/(1-z^2): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 38)): seq(coeff(Gser, z, n), n = 0 .. 34);
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CROSSREFS
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Cf. A202838, A202839, A202840, A202841, A202843, A202844
Sequence in context: A202850 A120262 A202849 * A013326 A202973 A074663
Adjacent sequences: A202839 A202840 A202841 * A202843 A202844 A202845
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch, Dec 25 2011
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STATUS
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approved
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