A182886 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps. These are paths that start at (0,0), end on the horizontal axis and whose steps are of the following four kinds: a (1,0)-step with weight 1; a (1,0)-step with weight 2; a (1,1)-step with weight 2; a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.
1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 0, 6, 1, 3, 1, 0, 6, 12, 3, 4, 1, 6, 0, 24, 21, 6, 5, 1, 0, 30, 12, 60, 34, 10, 6, 1, 0, 30, 90, 60, 121, 52, 15, 7, 1, 20, 0, 180, 215, 76, 21, 8, 1, 0, 140, 90, 630, 540, 421, 351, 107, 28, 9, 1, 0, 140, 560, 630, 1710, 1176, 846, 539, 146, 36, 10, 1
Offset: 0
Examples
T(3,2)=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, namely hH and Hh, have exactly two (1,0)-steps. Triangle starts: 1; 0, 1; 0, 1, 1; 2, 0, 2, 1; 0, 6, 1, 3, 1; 0, 6, 12, 3, 4, 1;
Links
- 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.
Programs
-
Maple
G:=1/sqrt(1-2*t*z-2*t*z^2+t^2*z^2+2*t^2*z^3+t^2*z^4-4*z^3): Gser:=simplify(series(G,z=0,14)): 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
Formula
G.f.: G(t,z) = 1/sqrt(1 - 2tz - 2tz^2 + t^2*z^2 + 2t^2*z^3 + t^2*z^4 - 4z^3).
Comments