%N 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 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).
%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 Sum(k*T(n,k), k>=0) = A202839(n-2).
%F T(n,0) = A202842(n).
%F 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 The multivariate g.f. H(z, t, t, ...) 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 = tz^2 + tz^4 + tz^6 + ... .
%e 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 Triangle starts:
%p 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 Cf. A004148, A202838, A202839, A202840, A202842, A202843, A202844
%A _Emeric Deutsch_, Dec 25 2011