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A202841 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of length 2 (n>=0, k>=0).

%I A202841 #9 Mar 05 2020 12:32:46
%S A202841 1,1,1,2,4,7,1,14,3,31,6,66,16,142,43,316,104,3,708,256,14,1593,647,
%T A202841 43,3625,1610,138,8314,3990,430,1,19165,9944,1247,16,44433,24762,3552,
%U A202841 85,103557,61574,10040,331,242376,153270,27877,1225,569514,381718,76491,4272,9
%N A202841 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of length 2 (n>=0, k>=0).
%C A202841 For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
%C A202841 Sum of entries in row n is A004148 (the secondary structure numbers).
%H A202841 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 A202841 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 A202841 Sum(k*T(n,k), k>=0) = A202839(n-2).
%F A202841 T(n,0) = A202842(n).
%F A202841 G.f. G(t,z) satisfies G = 1 + zG + [f/(1 + f)]G(G-1), where f = (t-1)z^4 + z^2/(1-z^2).
%F A202841 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 A202841 Row 5 is 7,1: 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; only the last one has a stack of length 2.
%e A202841 Triangle starts:
%e A202841 1;
%e A202841 1;
%e A202841 1;
%e A202841 2;
%e A202841 4;
%e A202841 7,1;
%e A202841 14,3;
%e A202841 31,6;
%p A202841 f := (t-1)*z^4+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, 23)): for n from 0 to 19 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 19 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
%Y A202841 Cf. A004148, A202838, A202839, A202840, A202842, A202843, A202844
%K A202841 nonn,tabf
%O A202841 0,4
%A A202841 _Emeric Deutsch_, Dec 25 2011