OFFSET
0,5
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..floor((n-1)/2)} k*A128733(n,k), for n >= 1.
G.f.: z^2*g^2*(g-1)/(1-3*z+2*z^2-3*z^2*g^2), where g = 1 + z*g^2 + z*(g-1) = (1 - z - sqrt(1 - 6*z + 5*z^2))/(2*z).
a(n) ~ 5^(n-1/2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
Conjecture: +2*(3*n-20)*(n-3)*(n+1)*a(n) +(-39*n^3+362*n^2-739*n+300)*a(n-1) +8*(n-2) *(6*n^2-49*n+75) *a(n-2) -5*(n-2)*(n-3)*(3*n-17)*a(n-3)=0. - R. J. Mathar, Jun 17 2016
EXAMPLE
a(4)=8 because we have UDUUUD(LD), UUUD(LD)UD, UUDUUD(LD), UUUUD(LD)D, UUUDUD(LD), UUUUDD(LD), UUUUDL(LD) and UUUUD(LD)L (the LD's are shown between parentheses; the other 28 skew Dyck paths of semilength 4 contain no LD).
MAPLE
g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=z^2*g^2*(g-1)/(1-3*z+2*z^2-3*z^2*g^2): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..26);
MATHEMATICA
CoefficientList[Series[x^2*((1-x-Sqrt[1-6*x+5*x^2])/2/x)^2*((1-x-Sqrt[1-6*x+5*x^2])/2/x-1)/(1-3*x+2*x^2-3*x^2*((1-x-Sqrt[1-6*x+5*x^2])/2/x)^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
PROG
(PARI) x='x+O('x^50); concat([0, 0, 0], Vec(x^2*((1-x-sqrt(1-6*x+5*x^2))/2/x)^2*((1-x-sqrt(1-6*x+5*x^2))/2/x-1)/(1-3*x+2*x^2-3*x^2*((1-x-sqrt(1-6*x+5*x^2))/2/x)^2))) \\ G. C. Greubel, Mar 20 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Mar 31 2007
STATUS
approved