OFFSET
0,8
COMMENTS
Row n has 1+2*floor(n/2) terms. Row sums yield the Catalan numbers (A000108).
LINKS
Chao-Jen Wang, Applications of the Goulden-Jackson cluster method to counting Dyck paths by occurrences of subwords, Dissertation, Brandeis University, 2011.
FORMULA
G.f.: G=G(t,z) satisfies: z(1-z+tz)^2*G^2-(1-z+tz)(1+z-tz)G+1 = 0.
EXAMPLE
T(4,3)=2 because we have (UU)D(UU)(DDD) and (UUU)(DD)U(DD) (here U=(1,1) and D=(1,-1); the long ascents and the long descents are shown between parentheses).
Triangle starts:
1;
1;
1,0,1;
1,0,4;
1,0,10,2,1;
1,0,20,12,9;
1,0,35,42,47,6,1;
MAPLE
G:=(1+z-t*z-sqrt(t^2*z^2-2*t*z^2-2*t*z+z^2-2*z+1))/2/z/(1-z+t*z): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 12 do seq(coeff(P[n], t, j), j=0..2*floor(n/2)) od; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Feb 27 2007
STATUS
approved