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A182886 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps. 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. 2
1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 0, 6, 1, 3, 1, 0, 6, 12, 3, 4, 1, 6, 0, 24, 21, 6, 5, 1, 0, 30, 12, 60, 34, 10, 6, 1, 0, 30, 90, 60, 121, 52, 15, 7, 1, 20, 0, 180, 215, 76, 21, 8, 1, 0, 140, 90, 630, 540, 421, 351, 107, 28, 9, 1, 0, 140, 560, 630, 1710, 1176, 846, 539, 146, 36, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

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

T(3n,0)=binom(2n,n) = A000984(n).

T(n,0)=0 if n mod 3 > 0.

Sum(k*T(n,k), k=0..n)= A182887(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..75.

FORMULA

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

EXAMPLE

T(3,2)=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.

Triangle starts:

1;

0,1;

0,1,1;

2,0,2,1;

0,6,1,3,1;

0,6,12,3,4,1;

MAPLE

G:=1/sqrt(1-2*t*z-2*t*z^2+t^2*z^2+2*t^2*z^3+t^2*z^4-4*z^3): 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, A182887, A000984.

Sequence in context: A080844 A321428 A076626 * A108731 A235168 A060950

Adjacent sequences:  A182883 A182884 A182885 * A182887 A182888 A182889

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Dec 11 2010

STATUS

approved

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Last modified January 20 16:20 EST 2019. Contains 319335 sequences. (Running on oeis4.)