A202844 Number of secondary structures of size n having no stacks of length 3.

1, 1, 1, 2, 4, 8, 17, 36, 79, 179, 407, 935, 2173, 5089, 12005, 28500, 68022, 163154, 393060, 950652, 2307454, 5618906, 13723145, 33607242, 82507764, 203028034, 500659653, 1237053269, 3062204227, 7593229687, 18858944533, 46909741893, 116848688876, 291449697298

Offset

0,4

Comments

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

Links

Table of n, a(n) for n=0..33.

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=G(z) satisfies G = 1+zG +fG(G-1)/(1+f), where f = z^2*(1-z^4+z^6)/(1-z^2).

a(n) = A202843(n,0).

Example

a(5)=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.

Maple

f := z^2*(1-z^4+z^6)/(1-z^2): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 37)): seq(coeff(Gser, z, n), n = 0 .. 33);

Crossrefs

Cf. A202838, A202839, A202840, A202841, A202842, A202843.

Sequence in context: A002845 A072925 A002955 * A093951 A137255 A247298

Adjacent sequences: A202841 A202842 A202843 * A202845 A202846 A202847

Keyword

nonn

Author

Emeric Deutsch, Dec 25 2011

Status

approved