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A110335 Number of valleys (i.e., (1,-1) followed by (1,1)) at level zero in all peakless Motzkin paths of length n+6 (can be easily translated into RNA secondary structure terminology). 1
1, 4, 12, 34, 92, 242, 627, 1608, 4096, 10388, 26269, 66304, 167161, 421162, 1060816, 2671908, 6730941, 16961430, 42758695, 107843080, 272136858, 687106696, 1735849310, 4387895300, 11098372185, 28088028612, 71128006458, 180224822694 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..27.

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

a(n) = Sum_{k>=0} k*A110333(n+6,k).

G.f.: 8/((1 - z + z^2 + Q)^2*(1 - 2z - z^2 + z^4 + (1 - z - z^2)Q)), where Q = sqrt(1 - 2z - z^2 - 2z^3 + z^4).

EXAMPLE

a(1)=4 because among the 37 (=A004148(7)) peakless Motzkin paths of length 7 only HUH(DU)HD, UH(DU)HDH, UH(DU)HHD and UHH(DU)HD have valleys at level zero (shown between parentheses; here U=(1,1), H=(1,0), D=(1,-1)).

MAPLE

Q:=sqrt(1-2*z-z^2-2*z^3+z^4): G:=8/(1-z+z^2+Q)^2/(1-2*z-z^2+z^4+(1-z-z^2)*Q): Gser:=series(G, z=0, 34): 1, seq(coeff(Gser, z^n), n=1..31);

CROSSREFS

Cf. A004148, A110333.

Sequence in context: A036880 A107069 A191823 * A166294 A307305 A176753

Adjacent sequences:  A110332 A110333 A110334 * A110336 A110337 A110338

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jul 20 2005

STATUS

approved

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Last modified May 25 11:27 EDT 2020. Contains 334592 sequences. (Running on oeis4.)