OFFSET
0,7
COMMENTS
REFERENCES
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.
FORMULA
G.f.: G(t,z) =1/sqrt(1-2tz-2z^2+t^2*z^2+2t*z^3+z^4-4z^3).
EXAMPLE
T(3,1)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them have exactly one h step.
Triangle starts:
1;
0,1;
1,0,1;
2,2,0,1;
1,6,3,0,1;
6,3,12,4,0,1
MAPLE
G:=1/sqrt(1-2*t*z-2*z^2+t^2*z^2+2*t*z^3+z^4-4*z^3): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 11 do seq(coeff(P[n], t, k), k=0..n) od; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Dec 11 2010
STATUS
approved