login
A128732
Number DL's in all skew Dyck paths of semilength n.
2
0, 0, 1, 5, 23, 106, 493, 2312, 10917, 51840, 247319, 1184557, 5692517, 27434578, 132547877, 641789941, 3113487683, 15130119784, 73637665027, 358883327591, 1751237017413, 8555108199294, 41836182269267, 204779733440086
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 the path is defined to be the number of its steps.
LINKS
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
FORMULA
a(n) = Sum_{k>=0} k*A128731(n,k).
G.f.: z*(1 - z - sqrt(1 - 6*z + 5*z^2))/(1 - 6*z + 5*z^2 +(1+z)*sqrt(1 - 6*z + 5*z^2)).
a(n) ~ 5^(n-1/2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
Conjecture: +2*n*(3*n-1)*a(n) -n*(39*n-37)*a(n-1) +4*(12*n^2-22*n-15)*a(n-2) -5*(3*n+2)*(n-3)*a(n-3)=0. - R. J. Mathar, Jun 17 2016
EXAMPLE
a(3)=5 because we have UDUUDL, UUUDLD, UUDUDL, UUUDDL and UUUDLL (the remaining 5 paths are Dyck paths which, obviously, contain no DL's).
MAPLE
G:=z*(1-z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2+(1+z)*sqrt(1-6*z+5*z^2)): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..26);
MATHEMATICA
CoefficientList[Series[x*(1-x-Sqrt[1-6*x+5*x^2])/(1-6*x+5*x^2+(1+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(z*(1-z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2 +(1+z)*sqrt(1-6*z+5*z^2)))) \\ G. C. Greubel, Mar 20 2017
CROSSREFS
Cf. A128731.
Sequence in context: A239406 A107839 A270530 * A026894 A126473 A238112
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Mar 31 2007
STATUS
approved