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A098056 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k subwords of the type uh^ju, dH^jd, or dh^ju for some j>0, where u=(1,1), d=(1,-1) and h=(1,0) (can be easily expressed using RNA secondary structure terminology). 2
1, 1, 1, 2, 4, 8, 15, 2, 27, 9, 1, 48, 29, 5, 84, 80, 21, 147, 198, 74, 4, 257, 463, 230, 27, 1, 451, 1033, 667, 125, 7, 796, 2235, 1811, 488, 43, 1413, 4727, 4694, 1676, 219, 6, 2526, 9828, 11700, 5317, 946, 54, 1, 4544, 20192, 28252, 15813, 3696, 326, 9, 8226, 41100 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row n

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

A098056(n,0)=A098057(n).

Sum(k*A098056(n,k),k>=0)=A187259(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.

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

LINKS

Table of n, a(n) for n=0..57.

M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux at problemes d'enumeration en biologie moleculaire, Sem. Loth. Comb. B08l (1984) 79-86.

FORMULA

G.f.=G=G(t, z) satisfies G = 1 + zG + z^2*[H + 2tzH/(1-z)+t^2*z^2*H/(1-z)^2+ z/(1-z)][G-(1-t)zH/(1-z)^2], where H=(1-z)^2*G-1+z.

The 4-variate g.f. G(t,s,v,z) of peakless Motzkin paths, where t, s, v mark subwords of the types uH^ju, dH^jd, dH^ju, respectively, and z marks length, satisfies the equation

G = 1+zG+z^2*[H + (t+s)zH/(1-z)+tsz^2*H/(1-z)^2+z/(1-z)][G-(1-v)zH/(1-z)^2],

where H = (1-z)[(1-z)G-1]. As special cases we get the current sequence A098056 and the sequences A097777 and A098083.

EXAMPLE

Triangle starts:

1;

1;

1;

2;

4;

8;

15,2;

27,9,1;

48,29,5;

84.80,21;

147,198,74,7;

It seems that the number r(n) of terms in row n>=3 is given by r(n)=n/2-1 if

n=2 (mod 4) and r(n)=2*round(n/4)-1 otherwise (here round(m) is the nearest integer to m).

T(7,1)=9 because we have h(uhu)hdd, (uhhu)hdd, (uhu)hhdd, (uhu)hddh, uh(dhu)hd and the reflections of the first four paths in a vertical axis; here u=(1,1), h=(1,0), d=(1,-1) and the pertinent subwords are shown between parentheses.

CROSSREFS

Cf. A041048, A098057, A187259.

Sequence in context: A118890 A118869 A118897 * A097100 A002954 A019278

Adjacent sequences:  A098053 A098054 A098055 * A098057 A098058 A098059

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Sep 11 2004r

STATUS

approved

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Last modified May 7 15:44 EDT 2021. Contains 343651 sequences. (Running on oeis4.)