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A202845
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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).
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3
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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
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OFFSET
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0,7
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COMMENTS
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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).
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LINKS
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FORMULA
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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 + ... .
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EXAMPLE
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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;
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MAPLE
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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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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