OFFSET
0,4
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 n contains 1 + floor(n/2) terms.
Row sums yield A002212.
LINKS
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
FORMULA
Sum_{k=0..floor(n/2)} k*T(n,k) = A129169(n).
G.f.: G(t,z) = (2 + t - 3tz - t*sqrt(1 - 6z + 5z^2))/(2 - t - 2z + 3tz + t*sqrt(1 - 6z + 5z^2)).
EXAMPLE
T(4,2)=2 because we have UUDDUUDD and UUDDUUDL.
Triangle starts:
1;
1;
1, 2;
1, 9;
1, 33, 2;
1, 119, 17;
1, 443, 97, 2;
MAPLE
G:=(2+t-3*t*z-t*sqrt(1-6*z+5*z^2))/(2-t-2*z+3*t*z+t*sqrt(1-6*z+5*z^2)): 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/2)) od; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Apr 05 2007
STATUS
approved