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A128730 Number of UDL's in all skew Dyck paths of semilength n. 3
0, 0, 1, 4, 16, 68, 301, 1366, 6301, 29400, 138355, 655424, 3121438, 14930540, 71675839, 345148892, 1666432816, 8064278288, 39103576699, 189949958332, 924163714216, 4502711570988, 21966152501239, 107284324830302 (list; graph; refs; listen; history; text; internal format)
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 steps in it.

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} k*A128728(n,k).

G.f.: 2*z^2/(1-6*z+5*z^2+(1+z)*sqrt(1-6*z+5*z^2)).

a(n) ~ 5^(n-1/2)/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014

D-finite with recurrence: +2*(n-1)*(3*n-8)*a(n) +(-39*n^2+161*n-148)*a(n-1) +(48*n^2-215*n+220)*a(n-2) -5*(3*n-5)*(n-3)*a(n-3)=0. - R. J. Mathar, Jun 17 2016

For n >= 2, a(n) = Sum_{k=1..n-1} binomial(n,k)*A014300(k). - Jianing Song, Apr 20 2019

EXAMPLE

a(3) = 4 because we have UDUUDL, UUUDLD, UUDUDL and UUUDLL (the other six skew Dyck paths of semilength are the five Dyck paths and UUUDDL.

MAPLE

G:=2*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[2*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(2*z^2/(1-6*z+5*z^2+(1+z)*sqrt(1-6*z+5*z^2)))) \\ G. C. Greubel, Mar 19 2017

CROSSREFS

Cf. A128728, A014300.

Sequence in context: A290912 A089979 A179191 * A151243 A006319 A202020

Adjacent sequences:  A128727 A128728 A128729 * A128731 A128732 A128733

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Mar 31 2007

STATUS

approved

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Last modified December 5 12:24 EST 2021. Contains 349557 sequences. (Running on oeis4.)