A202848 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).

1, 1, 1, 2, 4, 7, 1, 14, 3, 31, 6, 66, 16, 141, 44, 313, 107, 3, 702, 262, 14, 1577, 663, 43, 3581, 1654, 138, 8207, 4091, 436, 1, 18903, 10178, 1275, 16, 43770, 25339, 3638, 85, 101903, 62952, 10316, 331, 238282, 156495, 28743, 1228, 559322, 389374, 78979, 4320, 9

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) = A202846(n-2).

T(n,0) = A202849(n).

Links

Table of n, a(n) for n=0..51.

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 = (z^2 + t*z^4)/(1-z^4).

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 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.
Triangle starts:
1;
1;
1;
2;
4;
7,1;
14,3;
31,6;

Maple

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

Crossrefs

Cf. A004145, A202846, A023427, A202849,

Sequence in context: A098073 A118390 A247294 * A202841 A247290 A246183

Adjacent sequences: A202845 A202846 A202847 * A202849 A202850 A202851

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 26 2011

Status

approved