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A182889 Number of weighted lattice paths in L_n having no (1,0)-steps 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, 0, 1, 2, 3, 8, 17, 38, 89, 206, 485, 1152, 2751, 6614, 15983, 38798, 94569, 231342, 567771, 1397562, 3449285, 8533886, 21161001, 52579900, 130896887, 326440746, 815437967, 2040049514, 5111051473, 12822135138, 32207384995, 80995950182, 203917464635 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

a(n)=A182888(n,0).

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..32.

FORMULA

G.f.: G(z) =1/( z+sqrt((1+z+z^2)(1-3z+z^2)) ).

a(n) ~ sqrt(7*sqrt(5)-15) * ((3 + sqrt(5))/2)^(n+2) / (sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 06 2016

EXAMPLE

a(3)=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 ud and du, have no h steps at level 0.

MAPLE

G:=1/(z+sqrt((1+z+z^2)*(1-3*z+z^2))): Gser:=series(G, z=0, 35): seq(coeff(Gser, z, n), n=0..32);

MATHEMATICA

CoefficientList[Series[1/(x+Sqrt[(1+x+x^2)(1-3x+x^2)]), {x, 0, 40}], x] (* Harvey P. Dale, Jun 16 2013 *)

CROSSREFS

Cf. A182888.

Sequence in context: A267223 A292401 A132333 * A256169 A298405 A219788

Adjacent sequences:  A182886 A182887 A182888 * A182890 A182891 A182892

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Dec 11 2010

STATUS

approved

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Last modified February 19 10:18 EST 2019. Contains 320310 sequences. (Running on oeis4.)