A182888 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps at level 0. These are paths of weight n that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 4, 3, 0, 1, 8, 7, 6, 4, 0, 1, 17, 20, 12, 8, 5, 0, 1, 38, 44, 36, 18, 10, 6, 0, 1, 89, 104, 82, 56, 25, 12, 7, 0, 1, 206, 253, 204, 132, 80, 33, 14, 8, 0, 1, 485, 604, 513, 344, 195, 108, 42, 16, 9, 0, 1, 1152, 1466, 1262, 891, 530, 272, 140, 52, 18, 10, 0, 1

Offset

0,7

Links

Table of n, a(n) for n=0..77.

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 and 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/( z-tz+sqrt((1+z+z^2)(1-3z+z^2)) ).

Sum_{k=0..n} k*T(n,k) = A182890(n).

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, namely hH and Hh, have exactly two (1,0)-steps at level 0.
Triangle starts:
   1;
   0,   1;
   1,   0,  1;
   2,   2,  0,  1;
   3,   4,  3,  0,  1;
   8,   7,  6,  4,  0,  1;
  17,  20, 12,  8,  5,  0, 1;
  38,  44, 36, 18, 10,  6, 0, 1;
  89, 104, 82, 56, 25, 12, 7, 0, 1;
  ...

Maple

G:=1/(z-t*z+sqrt((1+z+z^2)*(1-3*z+z^2))): 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

Crossrefs

Row sums give A051286.

Column k=0 gives A182889.

Cf. A182890.

Sequence in context: A086460 A321884 A136431 * A317205 A384899 A384900

Adjacent sequences: A182885 A182886 A182887 * A182889 A182890 A182891

Keyword

nonn,tabl

Author

Emeric Deutsch, Dec 11 2010

Status

approved