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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

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

T(n,0)=A182889(n).

Sum(k*T(n,k), k=0..n)=A182890(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..77.

FORMULA

G.f.: G(t,z) =1/( z-tz+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 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.

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

Cf. A051286, A182889, A182890.

Sequence in context: A124394 A086460 A136431 * A317205 A296068 A144064

Adjacent sequences:  A182885 A182886 A182887 * A182889 A182890 A182891

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Dec 11 2010

STATUS

approved

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Last modified October 22 11:11 EDT 2018. Contains 316438 sequences. (Running on oeis4.)