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A182891 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 at level 0. 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. 1
1, 1, 1, 1, 3, 2, 7, 3, 1, 15, 8, 3, 35, 21, 6, 1, 83, 50, 16, 4, 197, 123, 45, 10, 1, 473, 308, 117, 28, 5, 1145, 769, 304, 83, 15, 1, 2787, 1926, 798, 232, 45, 6, 6819, 4843, 2085, 636, 140, 21, 1, 16759, 12204, 5433, 1744, 416, 68, 7, 41345, 30813, 14154, 4749, 1200, 222, 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)=A182892(n).

Sum(k*T(n,k), k=0..n)=A182890(n-1).

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/[z^2-tz^2+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 one H-step at level 0.

Triangle starts:

1;

1;

1,1;

3,2;

7,3,1;

15,8,3;

35,21,6,1;

MAPLE

G:=1/(z^2-t*z^2+sqrt((1+z+z^2)*(1-3*z+z^2))): 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, A182890, A182892.

Sequence in context: A296513 A099378 A182885 * A071190 A295314 A057020

Adjacent sequences:  A182888 A182889 A182890 * A182892 A182893 A182894

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Dec 12 2010

STATUS

approved

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Last modified August 10 16:56 EDT 2020. Contains 336381 sequences. (Running on oeis4.)