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.
LINKS
G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 0..1000(terms 1..300 from Vincenzo Librandi)
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*A129170(n,k).
G.f.: z*(3-3*z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2)/2. - corrected by Vaclav Kotesovec, Oct 20 2012
Recurrence: (n-1)*a(n) = (11*n-19)*a(n-1) - 5*(7*n-17)*a(n-2) + 25*(n-3)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 3*5^(n-1)/2*(1-sqrt(5)/(6*sqrt(Pi*n))) . - Vaclav Kotesovec, Oct 20 2012
EXAMPLE
a(2)=6 because in the 3 skew Dyck paths of semilength 2, namely UDUD, UUDD and UUDL, the heights of the peaks are 1,1,2 and 2.
MAPLE
G:=z*(3-3*z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2)/2: Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27);
MATHEMATICA
CoefficientList[Series[x*(3 - 3*x - Sqrt[1 - 6*x + 5*x^2])/(1 - 6*x + 5*x^2)/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
PROG
(PARI) z='z+O('z^25); concat([0], Vec(z*(3-3*z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2)/2)) \\ G. C. Greubel, Feb 10 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 07 2007
STATUS
approved