A128730 - OEIS

%I #24 Sep 20 2024 05:43:09

%S 0,0,1,4,16,68,301,1366,6301,29400,138355,655424,3121438,14930540,

%T 71675839,345148892,1666432816,8064278288,39103576699,189949958332,

%U 924163714216,4502711570988,21966152501239,107284324830302

%N Number of UDL's in all skew Dyck paths of semilength n.

%C 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.

%H G. C. Greubel, <a href="/A128730/b128730.txt">Table of n, a(n) for n = 0..1000</a>

%H E. Deutsch, E. Munarini, S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

%F a(n) = Sum_{k>=0} k*A128728(n,k).

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

%F a(n) ~ 5^(n-1/2)/(6*sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 20 2014

%F 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

%F For n >= 2, a(n) = Sum_{k=1..n-1} binomial(n,k)*A014300(k). - _Jianing Song_, Apr 20 2019

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

%p 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);

%t 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 *)

%o (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

%Y Cf. A128728, A014300.

%K nonn

%O 0,4

%A _Emeric Deutsch_, Mar 31 2007