This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A128729 #14 Jul 22 2022 13:13:40 %S A128729 1,1,2,6,20,71,262,994,3852,15183,60686,245412,1002344,4129012, %T A128729 17135432,71575350,300690836,1269662127,5385593406,22938095326, %U A128729 98059308676,420610907183,1809690341366,7808145901068,33776362530776 %N A128729 Number of skew Dyck paths of semilength n with no UDL's. %C A128729 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 A128729 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 A128729 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 A128729 a(n) = A128728(n,0). %F A128729 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 A128729 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 A128729 a(2)=2 because we have UDUD and UUDD (UUDL does not qualify). %p A128729 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 A128729 Cf. A128728. %K A128729 nonn %O A128729 0,3 %A A128729 _Emeric Deutsch_, Mar 31 2007