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A129155 Number of skew Dyck paths of semilength n that have no primitive Dyck factors. 2
1, 0, 1, 4, 15, 59, 241, 1011, 4326, 18797, 82685, 367410, 1646494, 7432270, 33761322, 154213566, 707882503, 3263713148, 15107319268, 70182332975, 327111450097, 1529226524057, 7168880978609, 33693179852563 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
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. A primitive Dyck factor is a subpath of the form UPD that starts on the x-axis, P being a Dyck path.
LINKS
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
FORMULA
a(n) = A129154(n,0).
G.f.: (3-3*z-sqrt(1-6*z+5*z^2))/(2+z-sqrt(1-4*z)+sqrt(1-6*z+5*z^2)).
a(n) ~ (475 + 697*sqrt(5)) * 5^n / (3364*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014
EXAMPLE
a(3)=4 because we have UUUDLD, UUDUDL, UUUDDL and UUUDLL.
MAPLE
G:=(3-3*z-sqrt(1-6*z+5*z^2))/(2+z-sqrt(1-4*z)+sqrt(1-6*z+5*z^2)): Gser:=series(G, z=0, 32): seq(coeff(Gser, z, n), n=0..28);
MATHEMATICA
CoefficientList[Series[(3-3*x-Sqrt[1-6*x+5*x^2])/(2+x-Sqrt[1-4*x]+Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
PROG
(PARI) z='z+O('z^50); Vec((3-3*z-sqrt(1-6*z+5*z^2))/(2+z-sqrt(1-4*z)+sqrt(1-6*z+5*z^2))) \\ G. C. Greubel, Mar 20 2017
CROSSREFS
Sequence in context: A017951 A326212 A199210 * A219312 A271752 A291244
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 02 2007
STATUS
approved

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Last modified June 13 23:50 EDT 2024. Contains 373391 sequences. (Running on oeis4.)