A182885 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. These are paths 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; 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, 13, 10, 3, 27, 29, 6, 1, 61, 66, 22, 4, 133, 157, 75, 10, 1, 287, 398, 201, 40, 5, 633, 975, 538, 155, 15, 1, 1407, 2334, 1506, 476, 65, 6, 3121, 5631, 4077, 1414, 280, 21, 1, 6943, 13602, 10695, 4320, 966, 98, 7, 15517, 32623, 27966, 12765, 3150, 462, 28, 1

Offset

0,5

Comments

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

T(n,0)=A098479(n).

Sum(k*T(n,k), k=0..n)=A182886(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..63.

Formula

G.f.: G(t,z) =1/sqrt(1-2z-2tz^2+z^2+2t*z^3+t^2*z^4-4z^3).

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 have exactly one H step.
Triangle starts:
1;
1;
1,1;
3,2;
7,3,1;
13,10,3;
27,29,6,1;

Maple

G:=1/sqrt(1-2*z-2*t*z^2+z^2+2*t*z^3+t^2*z^4-4*z^3): 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, A098479, A182886.

Sequence in context: A355934 A296513 A099378 * A182891 A071190 A295314

Adjacent sequences: A182882 A182883 A182884 * A182886 A182887 A182888

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 11 2010

Status

approved