A202838 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).

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, 85, 65, 253, 356, 295, 9, 117, 494, 762, 805, 105, 214, 938, 1713, 1912, 594, 2, 391, 1783, 3828, 4326, 2331, 76, 708, 3456, 8265, 9882, 7290, 771, 1278, 6793, 17309, 23109, 19784, 4529, 30

Offset

0,7

Comments

For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.

Sum of entries in row n is A004148 (the secondary structure numbers).

Sum(k*T(n,k), k>=0)=A202839(n).

T(n,0)=A202840(n).

Links

Table of n, a(n) for n=0..61.

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.

P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

Formula

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).

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 + ... .

Example

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.
Triangle starts:
1;
1;
1;
1,1;
1,3;
2,6;
4,10,3;
8,15,14;

Maple

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

Crossrefs

Cf. A202839, A202840, A202841, A202842, A202843, A202844

Sequence in context: A257910 A006368 A202845 * A105354 A094077 A375113

Adjacent sequences: A202835 A202836 A202837 * A202839 A202840 A202841

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 25 2011

Status

approved