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 a path is defined to be the number of its steps.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
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-2} k*A128719(n,k) (n >= 2).
G.f.: (2zg - g + 1 - z + z^2)/(2zg + z - 1), where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).
Recurrence: 2*(n+1)*(121*n-348)*a(n) = (1663*n^2 - 4620*n + 1392)*a(n-1) - (2476*n^2 - 11133*n + 11787)*a(n-2) + 5*(n-4)*(211*n-537)*a(n-3). - Vaclav Kotesovec, Nov 19 2012
a(n) ~ 9*5^(n-3/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 19 2012
EXAMPLE
a(3)=4 because each of the paths UUUDDD, UUUDLD, UUUDDL and UUUDLL contains one UUU, while the other six paths of semilength 3 contain no UUU's.
MAPLE
G:=(1-5*z+4*z^2-2*z^3-(1-2*z)*sqrt(1-6*z+5*z^2))/2/z/sqrt(1-6*z+5*z^2): Gser:=series(G, z=0, 28): seq(coeff(Gser, z, n), n=0..25);
MATHEMATICA
CoefficientList[Series[(2*x*(1-x-Sqrt[1-6*x+5*x^2])/(2*x)-(1-x-Sqrt[1-6*x+5*x^2])/(2*x)+1-x+x^2)/(2*x*(1-x-Sqrt[1-6*x+5*x^2])/(2*x)+x-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 19 2012 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Mar 30 2007
STATUS
approved