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A202845 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of odd length (n>=0, k>=0).

%I A202845 #13 Mar 05 2020 11:08:48
%S A202845 1,1,1,1,1,1,3,2,6,4,10,3,7,16,14,11,30,40,1,17,62,90,16,28,126,184,
%T A202845 85,49,241,384,295,9,87,444,839,808,105,152,820,1845,1960,594,2,262,
%U A202845 1547,3938,4581,2331,76,453,2957,8134,10731,7326,771,794,5636,16529,25110,20204,4529,30
%N A202845 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of odd length (n>=0, k>=0).
%C A202845 For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
%C A202845 Sum of entries in row n is A004148 (the secondary structure numbers).
%C A202845 Sum(k*T(n,k), k>=0)=A202846(n).
%C A202845 T(n,0)=A023427(n).
%H A202845 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 A202845 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 A202845 G.f.: G(t,z) satisfies G = 1 + zG + [f/(1 + f)]G(G-1), where f = (tz^2 + z^4)/(1-z^4).
%F A202845 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 A202845 Row 5 is 2,6: representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, ABVBA, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv; except for the first two, each has 1 stack of length 1.
%e A202845 Triangle starts:
%e A202845 1;
%e A202845 1;
%e A202845 1;
%e A202845 1,1;
%e A202845 1,3;
%e A202845 2,6;
%e A202845 4,10,3;
%e A202845 7,16,14;
%p A202845 f := (t*z^2+z^4)/(1-z^4): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
%Y A202845 Cf. A004148, A202846, A202848, A202849,
%K A202845 nonn,tabf
%O A202845 0,7
%A A202845 _Emeric Deutsch_, Dec 26 2011