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A110333 Triangle read by rows: T(n,k) (n,k>=0) = number of peakless Motzkin paths of length n having k valleys (i.e., (1,-1) followed by (1,1)) at level zero (can be easily translated into RNA secondary structure terminology). 2
1, 1, 1, 2, 4, 8, 16, 1, 33, 4, 70, 12, 152, 32, 1, 336, 82, 5, 754, 206, 18, 1714, 512, 56, 1, 3940, 1264, 163, 6, 9145, 3109, 456, 25, 21406, 7634, 1243, 88, 1, 50478, 18737, 3326, 284, 7, 119814, 46006, 8781, 868, 33, 286045, 113062, 22955, 2556, 129, 1, 686456 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row n (n >= 3) has floor(n/3) terms.

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

LINKS

Table of n, a(n) for n=0..54.

W. R. Schmitt and M. S. Waterman, Linear trees and RNA secondary structure, Discrete Appl. Math., 51, 317-323, 1994.

P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1978), 261-272.

M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumeration en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.

FORMULA

T(n,0) = A110334(n).

Sum_{k>=0} k*T(n,k) = A110335(n-6) for n >= 6, 0 otherwise.

G.f.: (1 + z^2*g - tz^2*g - z^2 + tz^2)/(1 - z - z^3*g - tz^2*g + tz^3*g + z^3 + tz^2 - tz^3), where g = 1 + zg + z^2*g(g-1) = (1 - z + z^2 - sqrt(1 - 2z - z^2 - 2z^3 + z^4))/(2z^2) is the g.f. of the RNA secondary structure numbers (A004148).

EXAMPLE

T(10,2)=5 because we have HUH(DU)H(DU)HD, UH(DU)H(DU)HDH, UHH(DU)H(DU)HD, UH(DU)HH(DU)HD and UH(DU)H(DU)HHD, where U=(1,1), H=(1,0), D=(1,-1) and the valleys at level zero are shown between parentheses.

Triangle begins:

    1;

    1;

    1;

    2;

    4;

    8;

   16,  1;

   33,  4;

   70, 12;

  152, 32,  1;

  336, 62,  5;

MAPLE

g:=(1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))/2/z^2: G:=(1+z^2*g-z^2*g*t-z^2+t*z^2)/(1-z-z^3*g-t*z^2*g+t*z^3*g+z^3+t*z^2-t*z^3): Gser:=simplify(series(G, z=0, 23)): P[0]:=1: for n from 1 to 20 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 2 do print(1) od: for n from 3 to 20 do seq(coeff(t*P[n], t^k), k=1..floor(n/3)) od; # yields sequence in triangular form

CROSSREFS

Cf. A004148, A110334, A110335.

Sequence in context: A317501 A097777 A089738 * A247292 A069783 A102251

Adjacent sequences:  A110330 A110331 A110332 * A110334 A110335 A110336

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Jul 20 2005

EXTENSIONS

Keyword tabf added by Michel Marcus, Apr 09 2013

STATUS

approved

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Last modified August 4 12:54 EDT 2020. Contains 336201 sequences. (Running on oeis4.)