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 ceiling(n/2) terms (n >= 1).
Row sums yield A002212.
Apparently a(n) = A126177(n-1). - Georg Fischer, Oct 28 2018
LINKS
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
FORMULA
EXAMPLE
T(5,2)=2 because we have UU(DDU)U(DDU)D and UUU(DDU)(DDU)D (the 2 subwords are shown between parentheses).
Triangle starts:
1;
1;
3;
9, 1;
27, 9;
81, 54, 2;
243, 270, 30;
729, 1215, 270, 5;
MAPLE
T:=(n, k)->3^(n-1-2*k)*binomial(n, k)*binomial(n-k, k+1)/n: 1; for n from 1 to 13 do seq(T(n, k), k=0..floor((n-1)/2)) od; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Mar 31 2007
STATUS
approved