|
| |
|
|
A114710
|
|
Number of hill-free Schroeder paths of length 2n that have no horizontal steps on the x-axis (0<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A hill is a peak at height 1.
|
|
1
| |
|
|
1, 0, 2, 6, 26, 114, 526, 2502, 12194, 60570, 305526, 1560798, 8058714, 41987106, 220470942, 1165553718, 6198683090, 33140219946, 178012804678, 960232902606, 5199384505226, 28250295397170, 153977094874862, 841656387060006
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Column 0 of A114709.
Hankel transform is 2^C(n+1,2) (A006125(n+1)). Hankel transform of a(n+1) is (2-2^(n+1))*2^C(n+1,2). [From Paul Barry (pbarry(AT)wit.ie), Oct 31 2008]
|
|
|
FORMULA
| G.f.=2/[1+3z+sqrt(1-6z+z^2)].
|
|
|
EXAMPLE
| a(3)=6 because we have UHHD, UHUDD, UUDHD, UUDUDD, UUHDD and UUUDDD.
|
|
|
MAPLE
| G:=2/(1+3*z+sqrt(1-6*z+z^2)): Gser:=series(G, z=0, 32): 1, seq(coeff(Gser, z^n), n=1..27);
|
|
|
CROSSREFS
| Cf. A114709.
Sequence in context: A191821 A192403 A050890 * A092880 A192808 A034474
Adjacent sequences: A114707 A114708 A114709 * A114711 A114712 A114713
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 26 2005
|
| |
|
|
