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.
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) = Sum_{k>=0} k*A128731(n,k).
G.f.: z*(1 - z - sqrt(1 - 6*z + 5*z^2))/(1 - 6*z + 5*z^2 +(1+z)*sqrt(1 - 6*z + 5*z^2)).
a(n) ~ 5^(n-1/2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
Conjecture: +2*n*(3*n-1)*a(n) -n*(39*n-37)*a(n-1) +4*(12*n^2-22*n-15)*a(n-2) -5*(3*n+2)*(n-3)*a(n-3)=0. - R. J. Mathar, Jun 17 2016
EXAMPLE
a(3)=5 because we have UDUUDL, UUUDLD, UUDUDL, UUUDDL and UUUDLL (the remaining 5 paths are Dyck paths which, obviously, contain no DL's).
MAPLE
G:=z*(1-z-sqrt(1-6*z+5*z^2))/(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..26);
MATHEMATICA
CoefficientList[Series[x*(1-x-Sqrt[1-6*x+5*x^2])/(1-6*x+5*x^2+(1+x)*Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
PROG
(PARI) z='z+O('z^50); concat([0, 0], Vec(z*(1-z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2 +(1+z)*sqrt(1-6*z+5*z^2)))) \\ G. C. Greubel, Mar 20 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Mar 31 2007
STATUS
approved