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%I #17 Jul 26 2022 16:02:51
%S 0,0,0,1,3,6,16,43,110,284,733,1886,4853,12486,32121,82647,212699,
%T 547552,1410023,3632260,9360140,24129284,62224692,160522287,414246823,
%U 1069376386,2761502201,7133442743,18432633823,47643696626,123182434292,318575889057,824125660356
%N Number of stacks of length 1 in all 2ndary structures of size n.
%C For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
%C Number of stacks of length 2 in all 2ndary structures of size n+2.
%C Number of stacks of length 3 in all 2ndary structures of size n+4.
%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 a(n) = Sum(k*A202838(n,k), k>=0).
%F a(n) = Sum(k*A202841(n+2,k), k>=0).
%F a(n) = Sum(k*A202843(n+4,k), k>=0).
%F G.f.: g(z) = z^2*(1-z^2)^2*S(S - 1)/(1 - z + z^2 -2*z^2*S), where S is defined by S = 1 + z*S + z^2*S(S-1) (the g.f. of the secondary structure numbers A004148).
%F a(n) ~ 5^(3/4) * phi^(2*n-3) / (2*sqrt(Pi*n)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, May 29 2022
%F D-finite with recurrence -(n+2)*(406*n-3981)*a(n) +(2022*n^2-15917*n-13552)*a(n-1) +4*(-402*n^2+2594*n+593)*a(n-2) +4*(-605*n^2+7719*n-23415)*a(n-3) +4*(-203*n^2-527*n+15295)*a(n-4) +2*(804*n^2-8404*n+14555)*a(n-5) +(2826*n^2-42913*n+153174)*a(n-6) -(1210*n-6753)*(n-10)*a(n-7)=0. - _R. J. Mathar_, Jul 26 2022
%e a(5)=6: 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 g := z^2*(1-z^2)^2*S*(S-1)/(1-z+z^2-2*z^2*S): S := ((1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^2: gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);
%t CoefficientList[Series[-(1 - x^2)^2 * ((1 - x) + (-1 + 2*x + x^3) / Sqrt[(1 - 3*x + x^2) * (1 + x + x^2)]) / (2*x^2), {x, 0, 40}], x] (* _Vaclav Kotesovec_, May 29 2022 *)
%Y Cf. A202838, A202840, A202841, A202842, A202843, A202844
%K nonn
%O 0,5
%A _Emeric Deutsch_, Dec 25 2011
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