A202843 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of length 3.

1, 1, 1, 2, 4, 8, 17, 36, 1, 79, 3, 179, 6, 407, 16, 935, 43, 2173, 110, 5089, 284, 12005, 727, 3, 28500, 1858, 14, 68022, 4767, 43, 163154, 12210, 138, 393060, 31255, 433, 950652, 80057, 1295, 2307454, 205088, 3804, 1, 5618906, 525534, 10985, 16, 13723145, 1347174, 31297, 85

Offset

0,4

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-4).

T(n,0) = A202844(n).

Links

Table of n, a(n) for n=0..50.

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^6 + 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 8: 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; none of them has stacks of length 3.
Triangle starts:
1;
1;
1;
2;
4;
8;
17;
36,1;
79,3;

Maple

f := (t-1)*z^6+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, 26)): for n from 0 to 22 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 22 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form

Crossrefs

Cf. A004148, A202838, A202839, A202840, A202841, A202842, A202844

Sequence in context: A097107 A098083 A182900 * A247297 A292322 A008999

Adjacent sequences: A202840 A202841 A202842 * A202844 A202845 A202846

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 25 2011

Status

approved