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%I #12 Jul 29 2022 17:19:05
%S 1,1,2,5,14,41,124,386,1230,3992,13150,43856,147796,502530,1721856,
%T 5939353,20608102,71879003,251876040,886309559,3130552258,11095355269,
%U 39447022648,140645181280,502773092420,1801633916188,6470373097004
%N Number of skew Dyck paths of semilength n having no DD'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) = A128738(n,0).
%F G.f.: G = G(t,z) satisfies z^2*G^3 - z(1-z)G^2 - (1-z)(1-3z)G + (1-z)^2 = 0.
%F D-finite with recurrence 15*n*(n+1)*a(n) -n*(83*n-49)*a(n-1) +(127*n^2-263*n+90)*a(n-2) +3*(-43*n^2+117*n-20)*a(n-3) +10*(n+1)*(7*n-27)*a(n-4) +16*(n-5)*(4*n-21)*a(n-5) -64*(n-5)*(n-6)*a(n-6)=0. - _R. J. Mathar_, Jul 22 2022
%e a(3)=5 because we have UDUDUD, UDUUDL, UUUDLD, UUDUDL and UUUDLL.
%p eq:=z^2*G^3-z*(1-z)*G^2-(1-z)*(1-3*z)*G+(1-z)^2=0: G:=RootOf(eq,G): Gser:=series(G,z=0,35): seq(coeff(Gser,z,n),n=0..30);
%Y Cf. A128738.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Mar 31 2007