A182878 Triangle read by rows: T(n,k) is the number of lattice paths L_n of weight n having length k (0 <= k <= n). These are paths that start at (0,0) and 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, and a (1,-1)-step with weight 1.

1, 0, 1, 0, 1, 1, 0, 0, 4, 1, 0, 0, 1, 9, 1, 0, 0, 0, 9, 16, 1, 0, 0, 0, 1, 36, 25, 1, 0, 0, 0, 0, 16, 100, 36, 1, 0, 0, 0, 0, 1, 100, 225, 49, 1, 0, 0, 0, 0, 0, 25, 400, 441, 64, 1, 0, 0, 0, 0, 0, 1, 225, 1225, 784, 81, 1, 0, 0, 0, 0, 0, 0, 36, 1225, 3136, 1296, 100, 1, 0, 0, 0, 0, 0, 0, 1, 441, 4900, 7056, 2025, 121, 1

Offset: 0

Author

Emeric Deutsch, Dec 10 2010

Keywords

• nonn
• tabl

Comments

The weight of a path is the sum of the weights of its steps.

Sum of entries in row n is A051286(n).

Sum_{k=0..n} k*T(n,k) = A182879(n).

Examples

			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;
		

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.

Crossrefs

Cf. A051286, A182879.

Programs

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

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).

Extensions

Keyword tabl added by Michel Marcus, Apr 09 2013