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A202848 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of even length (n>=0, k>=0).

%I A202848 #11 Mar 05 2020 11:09:29
%S A202848 1,1,1,2,4,7,1,14,3,31,6,66,16,141,44,313,107,3,702,262,14,1577,663,
%T A202848 43,3581,1654,138,8207,4091,436,1,18903,10178,1275,16,43770,25339,
%U A202848 3638,85,101903,62952,10316,331,238282,156495,28743,1228,559322,389374,78979,4320,9
%N A202848 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of even length (n>=0, k>=0).
%C A202848 For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
%C A202848 Sum of entries in row n is A004148 (the secondary structure numbers).
%C A202848 Sum(k*T(n,k), k>=0) = A202846(n-2).
%C A202848 T(n,0) = A202849(n).
%H A202848 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 A202848 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 A202848 G.f.: G(t,z) satisfies G = 1 + zG + [f/(1 + f)]G(G-1), where f = (z^2 + t*z^4)/(1-z^4).
%F A202848 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 A202848 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; the last one has 1 stack of length 2.
%e A202848 Triangle starts:
%e A202848 1;
%e A202848 1;
%e A202848 1;
%e A202848 2;
%e A202848 4;
%e A202848 7,1;
%e A202848 14,3;
%e A202848 31,6;
%p A202848 f := (z^2+t*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, 25)): 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 A202848 Cf. A004145, A202846, A023427, A202849,
%K A202848 nonn,tabf
%O A202848 0,4
%A A202848 _Emeric Deutsch_, Dec 26 2011