OFFSET
0,3
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 a 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..n-1} k*A128718(n,k).
G.f.: (1-4*z+z^2+(z-1)*sqrt(1-6*z+5*z^2))/(2*z*sqrt(1-6*z+5*z^2)).
a(n) ~ 3*5^(n-1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
Conjecture: (n+1)*(n-2)^2*a(n) -(n-1)*(6*n^2-15*n+4)*a(n-1) +5*(n-2)*(n-1)^2*a(n-2)=0. - R. J. Mathar, Jun 17 2016
Conjecture verified using the differential equation 4*g(z)+(20*z^3+2*z^2-2*z)*g'(z)+(25*z^4-15*z^3)*g''(z)+(5*z^5-6*z^4+z^3)*g'''(z)=0 satisfied by the G.f. - Robert Israel, Dec 25 2017
EXAMPLE
a(2)=2 because the paths of semilength 2 are UDUD, UUDD and UUDL, having altogether 2 UU's.
MAPLE
G:=(1-4*z+z^2+(z-1)*sqrt(1-6*z+5*z^2))/2/z/sqrt(1-6*z+5*z^2): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..25);
MATHEMATICA
CoefficientList[Series[(1-4*x+x^2+(x-1)*Sqrt[1-6*x+5*x^2])/2/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((1-4*z+z^2+(z-1)*sqrt(1-6*z+5*z^2))/(2*z*sqrt(1-6*z+5*z^2)))) \\ G. C. Greubel, Mar 20 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Mar 30 2007
STATUS
approved