|
%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
|
