A202845 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of odd length (n>=0, k>=0).

1, 1, 1, 1, 1, 1, 3, 2, 6, 4, 10, 3, 7, 16, 14, 11, 30, 40, 1, 17, 62, 90, 16, 28, 126, 184, 85, 49, 241, 384, 295, 9, 87, 444, 839, 808, 105, 152, 820, 1845, 1960, 594, 2, 262, 1547, 3938, 4581, 2331, 76, 453, 2957, 8134, 10731, 7326, 771, 794, 5636, 16529, 25110, 20204, 4529, 30

Offset: 0

Author

Emeric Deutsch, Dec 26 2011

Keywords

• nonn
• tabf

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).

T(n,0)=A023427(n).

Examples

			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, ABVBA, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv; except for the first two, each has 1 stack of length 1.
Triangle starts:
1;
1;
1;
1,1;
1,3;
2,6;
4,10,3;
7,16,14;
		

Links

• 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.

Crossrefs

Cf. A004148, A202846, A202848, A202849,

Programs

Maple

f := (t*z^2+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, 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

Formula

G.f.: G(t,z) satisfies G = 1 + zG + [f/(1 + f)]G(G-1), where f = (tz^2 + 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 + ... .