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A098083 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k DHH...HU's, where U=(1,1), D=(1,-1) and H=(1,0) (can be easily expressed using RNA secondary structure terminology). 3
1, 1, 1, 2, 4, 8, 17, 36, 1, 77, 5, 167, 18, 365, 58, 805, 172, 1, 1790, 486, 7, 4008, 1331, 34, 9033, 3561, 141, 20477, 9370, 524, 1, 46663, 24350, 1810, 9, 106843, 62674, 5930, 55, 245691, 160126, 18652, 279, 567194, 406732, 56832, 1245, 1, 1314086, 1028360 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row n >= 3 has ceiling((n-2)/4) terms.

Row sums yield the RNA secondary structure numbers (A004148).

T(n,0) = A190162(n).

Sum_{k>=0} k*T(n,k) = A190163(n).

LINKS

Table of n, a(n) for n=0..49.

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.

M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Sem. Loth. Comb. B08l (1984) 79-86. [Formerly: Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, p. 79-86.]

FORMULA

G.f. = G = G(t, z) satisfies G = 1 + z*G + z^2*(G-1)*(G - (1-t)*z*(G-z*G-1)/(1-z)).

EXAMPLE

Triangle starts:

    1;

    1;

    1;

    2;

    4;

    8;

   17;

   36,  1;

   77,  5;

  167, 18

T(8,1)=5 because we have UH(DHU)HHD, HUH(DHU)HD, UH(DHHU)HD, UH(DHU)HDH and UHH(DHU)HD (the required subwords are shown between parentheses).

MAPLE

eq := G = 1+z*G+z^2*(G-1)*(G-(1-t)*z*(G-1-z*G)/(1-z)): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 23)): for n from 0 to 20 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 20 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form

CROSSREFS

Cf. A004148.

Sequence in context: A136750 A274115 A097107 * A182900 A202843 A247297

Adjacent sequences:  A098080 A098081 A098082 * A098084 A098085 A098086

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Sep 13 2004

STATUS

approved

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Last modified June 14 20:21 EDT 2021. Contains 345038 sequences. (Running on oeis4.)