

A129155


Number of skew Dyck paths of semilength n that have no primitive Dyck factors.


2



1, 0, 1, 4, 15, 59, 241, 1011, 4326, 18797, 82685, 367410, 1646494, 7432270, 33761322, 154213566, 707882503, 3263713148, 15107319268, 70182332975, 327111450097, 1529226524057, 7168880978609, 33693179852563
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OFFSET

0,4


COMMENTS

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the xaxis, 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 xaxis, P being a Dyck path.


LINKS

E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 21912203


FORMULA

G.f.: (33*zsqrt(16*z+5*z^2))/(2+zsqrt(14*z)+sqrt(16*z+5*z^2)).
a(n) ~ (475 + 697*sqrt(5)) * 5^n / (3364*sqrt(Pi)*n^(3/2)).  Vaclav Kotesovec, Mar 20 2014


EXAMPLE

a(3)=4 because we have UUUDLD, UUDUDL, UUUDDL and UUUDLL.


MAPLE

G:=(33*zsqrt(16*z+5*z^2))/(2+zsqrt(14*z)+sqrt(16*z+5*z^2)): Gser:=series(G, z=0, 32): seq(coeff(Gser, z, n), n=0..28);


MATHEMATICA

CoefficientList[Series[(33*xSqrt[16*x+5*x^2])/(2+xSqrt[14*x]+Sqrt[16*x+5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)


PROG

(PARI) z='z+O('z^50); Vec((33*zsqrt(16*z+5*z^2))/(2+zsqrt(14*z)+sqrt(16*z+5*z^2))) \\ G. C. Greubel, Mar 20 2017


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



