|
| |
|
|
A089741
|
|
Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k UHH...HD's, where U=(1,1), D=(1,-1) and H=(1,0) (can be easily expressed using RNA secondary structure terminology).
|
|
1
| |
|
|
1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 15, 1, 1, 31, 5, 1, 63, 18, 1, 127, 56, 1, 1, 255, 160, 7, 1, 511, 432, 34, 1, 1023, 1120, 138, 1, 1, 2047, 2816, 500, 9, 1, 4095, 6912, 1672, 55, 1, 8191, 16640, 5264, 275, 1, 1, 16383, 39424, 15808, 1205, 11, 1, 32767, 92160, 45696, 4797
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,7
|
|
|
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. S. Waterman, Home Page (contains copies of his papers)
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.=[1-2z+2z^2-tz^3-sqrt[(1-tz^3)*(1-4z+4z^2-tz^3)]]/[2z^2*(1-z].
|
|
|
EXAMPLE
| T(7,2)=5 because we have H(UHD)(UHD), (UHD)H(UHD), (UHD)(UHD)H, (UHD)(UHHD) and (UHHD)(UHD) (the required subwords are shown between parentheses).
1; 1; 1; 1,1; 1,3; 1,7; 1,15,1; 1,31,5; 1,63,18; 1,127,56,1; 1,255,160,7;
|
|
|
CROSSREFS
| Row sums yield A004148. Column 1 is A000225, column 2 is A001793.
Sequence in context: A098093 A160627 A114712 * A089736 A205479 A094024
Adjacent sequences: A089738 A089739 A089740 * A089742 A089743 A089744
|
|
|
KEYWORD
| nonn,tabf
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 08 2004
|
| |
|
|
