OFFSET
0,9
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
T(n,k) = binomial(n,n-k)^2.
G.f. = G(t,z) = ((1-t*z)^2 - 2*t*z^2 - 2*t^2*z^3 + t^2*z^4)^(-1/2).
EXAMPLE
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 hhh, hH, Hh, ud, and du, having lengths 3, 2, 2, 2, and 2, respectively.
Triangle starts:
1;
0, 1;
0, 1, 1;
0, 0, 4, 1;
0, 0, 1, 9, 1;
0, 0, 0, 9, 16, 1;
MAPLE
T:=(n, k)->binomial(k, n-k)^2: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Dec 10 2010
EXTENSIONS
Keyword tabl added by Michel Marcus, Apr 09 2013
STATUS
approved