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A202840 Number of secondary structures of size n having no stacks of length 1.

%I A202840 #11 Jul 26 2022 16:04:46
%S A202840 1,1,1,1,1,2,4,8,14,23,38,65,117,214,391,708,1278,2318,4238,7803,
%T A202840 14419,26684,49433,91736,170656,318280,594905,1113868,2088554,3921505,
%U A202840 7373367,13883045,26174600,49408932,93372078,176637791,334491586,634023965,1202894908,2284187117
%N A202840 Number of secondary structures of size n having no stacks of length 1.
%C A202840 For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
%C A202840 a(n) = A202838(n,0).
%H A202840 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 A202840 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 A202840 G.f. G=G(z) satisfies G = 1+zG +fG(G-1)/(1+f), where f = z^4/(1-z^2).
%F A202840 D-finite with recurrence +(n+4)*a(n) +(-2*n-5)*a(n-1) +(-n-1)*a(n-2) +2*(2*n-1)*a(n-3) +(-n+2)*a(n-4) +4*(-2*n+7)*a(n-5) +3*(n-5)*a(n-6) +3*(2*n-13)*a(n-7) +2*(-n+8)*a(n-8) +2*(-2*n+19)*a(n-9) +(n-11)*a(n-10)=0. - _R. J. Mathar_, Jul 26 2022
%e A202840 a(5)=2; 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; they have 0,1,1,1,1,1,1,0 stacks of length 1, respectively.
%p A202840 f := 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, 42)): seq(coeff(Gser, z, n), n = 0 .. 39);
%Y A202840 Cf. A202838, A202839, A202841, A202842, A202843, A202844
%K A202840 nonn
%O A202840 0,6
%A A202840 _Emeric Deutsch_, Dec 25 2011