%I #11 Jan 22 2022 16:51:01
%S 1,1,1,2,4,8,17,36,79,179,407,935,2173,5089,12005,28500,68022,163154,
%T 393060,950652,2307454,5618906,13723145,33607242,82507764,203028034,
%U 500659653,1237053269,3062204227,7593229687,18858944533,46909741893,116848688876,291449697298
%N Number of secondary structures of size n having no stacks of length 3.
%C For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
%H I. L. Hofacker, P. Schuster and P. F. Stadler, <a href="https://doi.org/10.1016/S0166-218X(98)00073-0">Combinatorics of RNA secondary structures</a>, Discrete Appl. Math., 88, 1998, 207-237.
%H P. R. Stein and M. S. Waterman, <a href="https://doi.org/10.1016/0012-365X(79)90033-5">On some new sequences generalizing the Catalan and Motzkin numbers</a>, Discrete Math., 26 (1979), 261-272.
%F G.f.: G=G(z) satisfies G = 1+zG +fG(G-1)/(1+f), where f = z^2*(1-z^4+z^6)/(1-z^2).
%F a(n) = A202843(n,0).
%e a(5)=8; 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; none of them has stacks of length 3.
%p f := z^2*(1-z^4+z^6)/(1-z^2): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 37)): seq(coeff(Gser, z, n), n = 0 .. 33);
%Y Cf. A202838, A202839, A202840, A202841, A202842, A202843.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Dec 25 2011
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