

A202843


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


6



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
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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(n4).


LINKS



FORMULA

G.f.: G(t,z) satisfies G = 1 + zG + [f/(1 + f)]G(G1), where f = (t1)z^6 + z^2/(1z^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(H1), 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 := (t1)*z^6+z^2/(1z^2): eq := G = 1+z*G+f*G*(G1)/(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



KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



