

A182903


Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k peaks.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. A peak is a (1,1)step followed by a (1,1)step.


1



1, 1, 2, 4, 1, 9, 2, 21, 5, 48, 14, 1, 112, 38, 3, 263, 104, 9, 623, 276, 31, 1, 1484, 730, 99, 4, 3550, 1921, 309, 14, 8525, 5034, 929, 56, 1, 20537, 13145, 2739, 205, 5, 49612, 34208, 7956, 716, 20, 120136, 88780, 22804, 2394, 90, 1, 291519, 229860, 64650
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OFFSET

0,3


COMMENTS

Number of entries in row n is 1+floor(n/3).
Sum of entries in row n is A051286(n).


REFERENCES

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163177.


LINKS



FORMULA

Let F=F(t,s,x,y,z) be the 5variate g.f. of the considered weighted lattice paths, where z marks weight, t (s) marks number of peaks (valleys), x (y) indicates that the path starts with a (1,1)step ((1,1)step). Then F(t,s,x,y,z)=1+z(1+z)F(t,s,1,1,z)+xz^3[t+H(t,s,z)1]F(t,s,s,1,z)+yz^3[s+H(s,t,z)1]F(t,s,1,t,z), where H=H(t,s,z) is given by H=1+zH+z^2*H+z^3*(t1+H)[s(H1zHz^2*H)+1+zH+z^2*H] (see A182900).


EXAMPLE

T(7,2)=3. Indeed, denoting by h (H) the (1,0)step of weight 1 (2), and U=(1,1), D=(1,1), we have hUDUD, UDhUD, UDUDh.
Triangle starts:
1;
1;
2;
4,1;
9,2;
21,5;
48,14,1;


CROSSREFS



KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



