A182891 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps of weight 2 at level 0. The members of L_n 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, 1, 1, 1, 3, 2, 7, 3, 1, 15, 8, 3, 35, 21, 6, 1, 83, 50, 16, 4, 197, 123, 45, 10, 1, 473, 308, 117, 28, 5, 1145, 769, 304, 83, 15, 1, 2787, 1926, 798, 232, 45, 6, 6819, 4843, 2085, 636, 140, 21, 1, 16759, 12204, 5433, 1744, 416, 68, 7, 41345, 30813, 14154, 4749, 1200, 222, 28, 1

Offset

0,5

Comments

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

T(n,0)=A182892(n).

Sum(k*T(n,k), k=0..n)=A182890(n-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.

Links

Table of n, a(n) for n=0..63.

Formula

G.f. G(t,z) =1/[z^2-tz^2+sqrt((1+z+z^2)(1-3z+z^2))].

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 one H-step at level 0.
Triangle starts:
1;
1;
1,1;
3,2;
7,3,1;
15,8,3;
35,21,6,1;

Maple

G:=1/(z^2-t*z^2+sqrt((1+z+z^2)*(1-3*z+z^2))): Gser:=simplify(series(G, z=0, 18)): for n from 0 to 14 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 14 do seq(coeff(P[n], t, k), k=0..floor(n/2)) od; # yields sequence in triangular form

Crossrefs

Cf. A051286, A182890, A182892.

Sequence in context: A296513 A099378 A182885 * A071190 A295314 A057020

Adjacent sequences: A182888 A182889 A182890 * A182892 A182893 A182894

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 12 2010

Status

approved