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Number of skew Dyck paths of semilength n and having no LDU's.
1

%I #12 Jul 22 2022 13:06:18

%S 1,1,3,10,35,127,474,1810,7043,27839,111503,451640,1847032,7616692,

%T 31637664,132252886,555967283,2348920207,9968617393,42477135370,

%U 181661283779,779492777031,3354893322350,14479454240492,62652100034380

%N Number of skew Dyck paths of semilength n and having no LDU's.

%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 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) = A128735(n,0).

%F G.f.: G = G(z) satisfies zG^3 = (1-2z)(G-1)(2G-1).

%F D-finite with recurrence 8*n*(n+1)*a(n) -4*n*(11*n-2)*a(n-1) +2*(-11*n^2+114*n-154)*a(n-2) +(61*n-153)*(5*n-16)*a(n-3) -4*(47*n-159)*(n-4)*a(n-4) -220*(n-4)*(n-5)*a(n-5)=0. - _R. J. Mathar_, Jul 22 2022

%e a(4)=35 because among the 36 (=A002212(4)) skew Dyck paths of semilength 4 only UUUDLDUD has a LDU.

%p eq:=z*G^3=(1-2*z)*(G-1)*(2*G-1): G:=RootOf(eq,G): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..27);

%Y Cf. A128735.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Mar 31 2007