OFFSET
1,2
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 the path is defined to be the number of its steps.
Row sums yield A002212.
T(n,1) = 1 + A002212(n-1) (indeed, the path U^nDL^(n-1) and the paths UDP, where P is a skew Dyck path of semilength n-1).
T(n,n) = binomial(2n-2,n-1)/n = A000108(n-1) (the Catalan numbers).
LINKS
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
FORMULA
EXAMPLE
T(3,2)=4 because we have UUDDUD, UUUDLD, UUDUDL and UUUDDL.
Triangle starts:
1;
2, 1;
4, 4, 2;
11, 9, 11, 5;
37, 21, 31, 34, 14;
MAPLE
g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: h:=(1-z-sqrt(z^2-2*z+1+4*t*z^2-4*t*z))/2/t/z: G:=t*z*h*g+z*(h-1): Gser:=simplify(series(G, z=0, 14)): for n from 1 to 11 do P[n]:=sort(expand(coeff(Gser, z, n))) od: for n from 1 to 11 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Apr 03 2007
STATUS
approved