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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. 0
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 (list; graph; refs; listen; history; text; internal format)
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: A245601 A296513 A099378 * A182891 A071190 A295314

Adjacent sequences:  A182882 A182883 A182884 * A182886 A182887 A182888

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Dec 11 2010

STATUS

approved

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Last modified August 4 05:13 EDT 2020. Contains 336201 sequences. (Running on oeis4.)