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A202839 Number of stacks of length 1 in all 2ndary structures of size n. 6


%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).

%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);

%Y Cf. A202838, A202840, A202841, A202842, A202843, A202844

%K nonn

%O 0,5

%A _Emeric Deutsch_, Dec 25 2011

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Last modified June 1 09:53 EDT 2020. Contains 334762 sequences. (Running on oeis4.)