|
| |
|
|
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; 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
| 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 (deutsch(AT)duke.poly.edu), Sep 11 2004r
|
| |
|
|
