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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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A098073 A118390 A247294 * A202841 A247290 A246183
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Dec 26 2011
STATUS
approved

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Last modified April 12 10:04 EDT 2024. Contains 371627 sequences. (Running on oeis4.)