A128732 - OEIS

%I #15 Jul 23 2017 12:16:35

%S 0,0,1,5,23,106,493,2312,10917,51840,247319,1184557,5692517,27434578,

%T 132547877,641789941,3113487683,15130119784,73637665027,358883327591,

%U 1751237017413,8555108199294,41836182269267,204779733440086

%N Number DL'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 the path is defined to be the number of its steps.

%H G. C. Greubel, <a href="/A128732/b128732.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*A128731(n,k).

%F 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)).

%F a(n) ~ 5^(n-1/2)/(3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 20 2014

%F 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

%e 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).

%p 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);

%t 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 *)

%o (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

%Y Cf. A128731.

%K nonn

%O 0,4

%A _Emeric Deutsch_, Mar 31 2007