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 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) = Sum_{k=0,..,n} k*A129154(n,k).
G.f.: (3-3*z-sqrt(1-6*z+5*z^2))*(1-sqrt(1-4*z))/(1 +z + sqrt(1 - 6*z + 5*z^2))^2.
a(n) ~ (5-sqrt(5)) * 5^(n+3/2) / (36*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014
EXAMPLE
a(2)=3 because in all skew Dyck paths of semilength 3, namely (UD)(UD), (UUDD) and UUDL, we have altogether 3 primitive Dyck factors (shown between parentheses).
MAPLE
G:=(3-3*z-sqrt(1-6*z+5*z^2))*(1-sqrt(1-4*z))/(1+z+sqrt(1-6*z+5*z^2))^2: Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27);
MATHEMATICA
CoefficientList[Series[(3-3*x-Sqrt[1-6*x+5*x^2])*(1-Sqrt[1-4*x])/ (1+x+Sqrt[1-6*x+5*x^2])^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
PROG
(PARI) z='z+O('z^25); concat([0], Vec((3-3*z-sqrt(1-6*z+5*z^2))*(1-sqrt(1-4*z))/(1 +z + sqrt(1 - 6*z + 5*z^2))^2)) \\ G. C. Greubel, Feb 09 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 02 2007
STATUS
approved