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A182894 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, 0, 2, 2, 4, 12, 24, 54, 130, 300, 706, 1686, 4028, 9686, 23426, 56866, 138584, 338940, 831508, 2045736, 5046240, 12477290, 30919122, 76774382, 190995224, 475979602, 1188125394, 2970282794, 7436232760, 18641883396, 46792219972, 117590713254 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

a(n)=A182893(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+z^2+sqrt((1+z+z^2)(1-3z+z^2)) ).

a(n) ~ sqrt(105 + 47*sqrt(5)) * ((3 + sqrt(5))/2)^n / (5*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 06 2016

Conjecture: n*a(n) +(-4*n+3)*a(n-1) +(n-3)*a(n-2) -3*a(n-3) +3*(5*n-14)*a(n-4) +6*(n-3)*a(n-5) +6*(n-4)*a(n-6) +4*(-n+6)*a(n-7)=0. - R. J. Mathar, Jun 14 2016

a(n) ~ phi^(2*n + 4) / (5^(3/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 23 2017

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 (1,0)-steps at level 0.

MAPLE

G:=1/(z+z^2+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+x^2+Sqrt[(1+x+x^2)*(1-3*x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2016 *)

CROSSREFS

Cf. A182893.

Sequence in context: A002840 A298477 A253677 * A286410 A285611 A288303

Adjacent sequences:  A182891 A182892 A182893 * A182895 A182896 A182897

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Dec 12 2010

STATUS

approved

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Last modified September 22 16:33 EDT 2020. Contains 337291 sequences. (Running on oeis4.)