|
| |
|
|
A110239
|
|
Number of (1,1) steps in all peakless Motzkin paths of length n (can be easily translated into RNA secondary structure terminology).
|
|
1
| |
|
|
1, 3, 8, 22, 58, 151, 392, 1013, 2612, 6728, 17318, 44564, 114671, 295099, 759576, 1955657, 5036741, 12976355, 33443190, 86221745, 222371926, 573713958, 1480677048, 3822708372, 9872424913, 25504336609, 65907869404, 170368399138
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 3,2
|
|
|
COMMENTS
| Row sums of A110238.
|
|
|
REFERENCES
| 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, 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.
|
|
|
FORMULA
| G.f.=z^2g^2*(g-1)/(1-z^2*g^2), 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
| a(5)=8 because in the 8 (=A004148(5)) peakless Motzkin paths of length 5, namely HHHHH, UHDHH, UHHDH, UHHHD, HUHDH, HUHHD, HHUHD and UUHDD (where U=(1,1), H=(1,0) and D=(1,-1)), we have alltogether 8 U steps.
|
|
|
MAPLE
| g:=(1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))/2/z^2: G:=z^2*g^2*(g-1)/(1-z^2*g^2): Gser:=series(G, z=0, 37): seq(coeff(Gser, z^n), n=3..34);
|
|
|
CROSSREFS
| Cf. A004148, A110238.
Sequence in context: A036882 A020962 A027243 * A001853 A003227 A077848
Adjacent sequences: A110236 A110237 A110238 * A110240 A110241 A110242
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 17 2005
|
| |
|
|
