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

0,9

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).

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.

Links

Table of n, a(n) for n=0..90.

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

Cf. A051286, A182879.

Sequence in context: A036877 A049763 A328290 * A221971 A378008 A297785

Adjacent sequences: A182875 A182876 A182877 * A182879 A182880 A182881

Keyword

nonn,tabl

Author

Emeric Deutsch, Dec 10 2010

Extensions

Keyword tabl added by Michel Marcus, Apr 09 2013

Status

approved