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
G. C. Greubel, Table of n, a(n) for n = 0..1000
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 02 2007
STATUS
approved