A128729 Number of skew Dyck paths of semilength n with no UDL's.

1, 1, 2, 6, 20, 71, 262, 994, 3852, 15183, 60686, 245412, 1002344, 4129012, 17135432, 71575350, 300690836, 1269662127, 5385593406, 22938095326, 98059308676, 420610907183, 1809690341366, 7808145901068, 33776362530776

Offset: 0

Author

Emeric Deutsch, Mar 31 2007

Keywords

• nonn

Comments

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.

Examples

			a(2)=2 because we have UDUD and UUDD (UUDL does not qualify).
		

Links

• E. Deutsch, E. Munarini, and S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
• Helmut Prodinger, Skew Dyck paths without up-down-left, arXiv:2203.10516 [math.CO], 2022.

Crossrefs

Cf. A128728.

Programs

Maple

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);

Formula

a(n) = A128728(n,0).

G.f.: G = G(z) satisfies z^2*G^3 - z(2-z)G^2 + (1 - z^2)G - 1 + z + z^2 = 0.

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