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 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 (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) = A202839(n-4). T(n,0) = A202844(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 = (t-1)z^6 + z^2/(1-z^2). The multivariate g.f. H(z, t, t, ...) 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 = tz^2 + tz^4 + tz^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 := (t-1)*z^6+z^2/(1-z^2): eq := G = 1+z*G+f*G*(G-1)/(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 Cf. A004148, A202838, A202839, A202840, A202841, A202842, A202844 Sequence in context: A097107 A098083 A182900 * A247297 A292322 A008999 Adjacent sequences:  A202840 A202841 A202842 * A202844 A202845 A202846 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Dec 25 2011 STATUS approved

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Last modified May 25 20:58 EDT 2020. Contains 334597 sequences. (Running on oeis4.)