A128728 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UDL's (n >= 0; 0 <= k <= floor((n+1)/2)).

1, 1, 2, 1, 6, 4, 20, 16, 71, 64, 2, 262, 261, 20, 994, 1084, 141, 3852, 4572, 854, 7, 15183, 19520, 4772, 112, 60686, 84139, 25416, 1128, 245412, 365404, 131270, 9120, 30, 1002344, 1596420, 664004, 64790, 660, 4129012, 7008544, 3309336, 422928

Offset

0,3

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.

Row n has 1 + floor((n+1)/3) terms.

Row sums yield A002212.

T(n,0) = A128729(n).

Sum_{k>=0} k*T(n,k) = A128730(n).

Apparently, T(3k-1,k) = binomial(3k-1,k)/(3k-1) = A006013(k-1).

Links

Table of n, a(n) for n=0..42.

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.

Yuxuan Zhang and Yan Zhuang, A subfamily of skew Dyck paths related to k-ary trees, arXiv:2306.15778 [math.CO], 2023.

Formula

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

Example

T(3,1)=4 because we have UDUUDL, UUUDLD, UUDUDL and UUUDLL.
Triangle starts:
    1;
    1;
    2,   1;
    6,   4;
   20,  16;
   71,  64,   2;
  262, 261,  20;

Maple

eq:=z^2*G^3-z*(2-z)*G^2+(1-z^2)*G-1+z+z^2-t*z^2=0: G:=RootOf(eq, G): Gser:=simplify(series(G, z=0, 17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 14 do seq(coeff(P[n], t, j), j=0..floor((n+1)/3)) od; # yields sequence in triangular form

Crossrefs

Cf. A002212, A006013, A128729, A128730.

Sequence in context: A357639 A005299 A185586 * A084950 A336382 A358282

Adjacent sequences: A128725 A128726 A128727 * A128729 A128730 A128731

Keyword

nonn,tabf

Author

Emeric Deutsch, Mar 31 2007

Status

approved