

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)).


3



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
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OFFSET

0,3


COMMENTS

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the xaxis, 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.
Apparently, T(3k1,k) = binomial(3k1,k)/(3k1) = A006013(k1).


LINKS

E. Deutsch, E. Munarini, and S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 21912203.


FORMULA

G.f.: G = G(t,z) satisfies z^2*G^3  z(2z)G^2 + (1z^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^3z*(2z)*G^2+(1z^2)*G1+z+z^2t*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



KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



