|
| |
|
|
A128714
|
|
Number of skew Dyck paths of semilength n ending with a left step. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.
|
|
3
| |
|
|
0, 0, 1, 4, 15, 58, 232, 954, 4010, 17156, 74469, 327168, 1452075, 6501156, 29326743, 133166064, 608188737, 2791992736, 12876049123, 59626721244, 277150709717, 1292583258866, 6046985696778, 28369001791034, 133436435891480
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,4
|
|
|
COMMENTS
| Number of skew Dyck paths of semilength n and ending with a down step is A033321(n).
|
|
|
LINKS
| E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
|
|
|
FORMULA
| G.f.=[1-3z-sqrt(1-6z+5z^2)]/[1+z+sqrt(1-6z+5z^2)]. G.f.=z(g-1)/(1-zg), where g=1+zg^2+z(g-1)=[1-z-sqrt(1-6z+5z^2)](2z).
|
|
|
EXAMPLE
| a(3)=4 because we have UDUUDL, UUDUDL, UUUDDL and UUUDLL.
|
|
|
MAPLE
| G:=(1-3*z-sqrt(1-6*z+5*z^2))/(1+z+sqrt(1-6*z+5*z^2)): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27);
|
|
|
CROSSREFS
| Cf. A033321.
Sequence in context: A003126 A160156 A102052 * A007342 A017951 A199210
Adjacent sequences: A128711 A128712 A128713 * A128715 A128716 A128717
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2007
|
| |
|
|
