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%I #9 Mar 05 2020 11:10:52
%S 1,1,1,2,4,8,17,36,1,79,3,179,6,407,16,935,43,2173,110,5089,284,12005,
%T 727,3,28500,1858,14,68022,4767,43,163154,12210,138,393060,31255,433,
%U 950652,80057,1295,2307454,205088,3804,1,5618906,525534,10985,16,13723145,1347174,31297,85
%N Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of length 3.
%C For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
%C Sum of entries in row n is A004148 (the secondary structure numbers).
%C Sum(k*T(n,k), k>=0) = A202839(n-4).
%C T(n,0) = A202844(n).
%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(t,z) satisfies G = 1 + zG + [f/(1 + f)]G(G-1), where f = (t-1)z^6 + z^2/(1-z^2).
%F The multivariate g.f. H(z, t[1], t[2], ...) of secondary structures with respect to size (marked by z) and number of stacks of length j (marked by t[j]) satisfies H = 1 + zH + [f/(1 + f)]H(H-1), where f = t[1]z^2 + t[2]z^4 + t[3]z^6 + ... .
%e Row 5 is 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.
%e Triangle starts:
%e 1;
%e 1;
%e 1;
%e 2;
%e 4;
%e 8;
%e 17;
%e 36,1;
%e 79,3;
%p f := (t-1)*z^6+z^2/(1-z^2): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 26)): for n from 0 to 22 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 22 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
%Y Cf. A004148, A202838, A202839, A202840, A202841, A202842, A202844
%K nonn,tabf
%O 0,4
%A _Emeric Deutsch_, Dec 25 2011
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