A202838 - OEIS

%I #10 Mar 05 2020 11:57:26

%S 1,1,1,1,1,1,3,2,6,4,10,3,8,15,14,14,27,40,1,23,56,90,16,38,122,178,

%T 85,65,253,356,295,9,117,494,762,805,105,214,938,1713,1912,594,2,391,

%U 1783,3828,4326,2331,76,708,3456,8265,9882,7290,771,1278,6793,17309,23109,19784,4529,30

%N Triangle read by rows: T(n,k) is the number of secondary structures of size n (n>=0) having k stacks of length 1 (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)=A202839(n).

%C T(n,0)=A202840(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 = (t-1)z^2 + z^2/(1-z^2).

%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 2,6: 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; they have 0,1,1,1,1,1,1,0 stacks of length 1, respectively.

%e Triangle starts:

%e 1;

%e 1;

%e 1;

%e 1,1;

%e 1,3;

%e 2,6;

%e 4,10,3;

%e 8,15,14;

%p f := (t-1)*z^2+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, 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 Cf. A202839, A202840, A202841, A202842, A202843, A202844

%K nonn,tabf

%O 0,7

%A _Emeric Deutsch_, Dec 25 2011