Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #14 Jul 22 2022 13:13:40
%S 1,1,2,6,20,71,262,994,3852,15183,60686,245412,1002344,4129012,
%T 17135432,71575350,300690836,1269662127,5385593406,22938095326,
%U 98059308676,420610907183,1809690341366,7808145901068,33776362530776
%N Number of skew Dyck paths of semilength n with no UDL'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 a path is defined to be the number of steps in it.
%H E. Deutsch, E. Munarini, and 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.
%H Helmut Prodinger, <a href="https://arxiv.org/abs/2203.10516">Skew Dyck paths without up-down-left</a>, arXiv:2203.10516 [math.CO], 2022.
%F a(n) = A128728(n,0).
%F G.f.: G = G(z) satisfies z^2*G^3 - z(2-z)G^2 + (1 - z^2)G - 1 + z + z^2 = 0.
%F D-finite with recurrence 4*n*(n+1)*a(n) -32*n*(n-1)*a(n-1) +3*(23*n^2-78*n+59)*a(n-2) -2*(n-3)*(10*n-47)*a(n-3) -44*(n-3)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jul 22 2022
%e a(2)=2 because we have UDUD and UUDD (UUDL does not qualify).
%p eq:=z^2*G^3-z*(2-z)*G^2+(1-z^2)*G-1+z+z^2=0: G:=RootOf(eq,G): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..27);
%Y Cf. A128728.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Mar 31 2007