A202848 - OEIS

%I #11 Mar 05 2020 11:09:29

%S 1,1,1,2,4,7,1,14,3,31,6,66,16,141,44,313,107,3,702,262,14,1577,663,

%T 43,3581,1654,138,8207,4091,436,1,18903,10178,1275,16,43770,25339,

%U 3638,85,101903,62952,10316,331,238282,156495,28743,1228,559322,389374,78979,4320,9

%N 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 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) = A202846(n-2).

%C T(n,0) = A202849(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 = (z^2 + t*z^4)/(1-z^4).

%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 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 Triangle starts:

%e 1;

%e 1;

%e 1;

%e 2;

%e 4;

%e 7,1;

%e 14,3;

%e 31,6;

%p 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 Cf. A004145, A202846, A023427, A202849,

%K nonn,tabf

%O 0,4

%A _Emeric Deutsch_, Dec 26 2011