

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).


3



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
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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) = A202846(n2).


LINKS



FORMULA

G.f.: G(t,z) satisfies G = 1 + zG + [f/(1 + f)]G(G1), where f = (z^2 + t*z^4)/(1z^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(H1), 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)/(1z^4): eq := G = 1+z*G+f*G*(G1)/(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



KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



