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 A202851 Triangle read by rows: T(n,k) is the number of secondary structures of size n with no stacks of length >=2 and having k stacks of length 1 (n>=0, k>=0). 0
 1, 1, 1, 1, 1, 1, 3, 1, 6, 1, 10, 3, 1, 15, 14, 1, 21, 40, 1, 1, 28, 90, 16, 1, 36, 175, 85, 1, 45, 308, 295, 9, 1, 55, 504, 805, 105, 1, 66, 780, 1876, 594, 2, 1, 78, 1155, 3906, 2331, 76, 1, 91, 1650, 7470, 7280, 771, 1, 105, 2288, 13365, 19404, 4529, 30, 1, 120, 3094, 22660, 45990, 19348, 650 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209. Sum of entries in row n is A202850(n). REFERENCES 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. LINKS FORMULA G.f. G(t,z) satisfies G = 1 + zG + [tz^2/(1 + tz^2)]G(G-1). 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 1,6: representing unpaired vertices by v and arcs by AA, BB, etc., the 7 (= A202850(5)) secondary structures of size 5 and with no stacks of length >=2 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv; with the exception of the first one, they all have 1 stack of length 1. Triangle starts: 1; 1; 1; 1,1; 1,3; 1,6; 1,10,3; 1,15,14; MAPLE eq := G = 1+z*G+t*z^2*G*(G-1)/(1+t*z^2): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 22)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 17 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form CROSSREFS Cf. A202850 Sequence in context: A317496 A304236 A145063 * A007650 A236540 A165552 Adjacent sequences:  A202848 A202849 A202850 * A202852 A202853 A202854 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Dec 27 2011 STATUS approved

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Last modified November 17 11:02 EST 2019. Contains 329226 sequences. (Running on oeis4.)