

A182887


Number of (1,0)steps in all weighted lattice paths in L_n. 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.


1



0, 1, 3, 7, 21, 60, 166, 463, 1281, 3521, 9645, 26322, 71606, 194283, 525897, 1420595, 3830445, 10311510, 27718028, 74410105, 199519155, 534400491, 1429944603, 3822761742, 10211093226, 27254110405, 72691102131, 193750155673, 516100470051
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OFFSET

0,3


COMMENTS

a(n)=A182884(n)+A182884(n1).
a(n)=Sum(k*A182886(n,k),k>=0).


REFERENCES

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163177.


LINKS

Table of n, a(n) for n=0..28.


FORMULA

G.f.: G=z(1+z)(1zz^2)/[(13z+z^2)(1+z+z^2)]^{3/2}.
a(n) ~ ((3 + sqrt(5))/2)^n * sqrt(n) / (2*5^(1/4)*sqrt(Pi)).  Vaclav Kotesovec, Mar 06 2016
Conjecture: +(n1)*(182*n279)*a(n) +(230*n^2+11*n+643) *a(n1) +(450*n^2+1603*n315) *a(n2) +(498*n^2+971*n+57) *a(n3) +(86*n^2+959*n529) *a(n4) +(134*n59)*(n3) *a(n5)=0.  R. J. Mathar, Jun 14 2016


EXAMPLE

a(3)=7. 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; the total number of (1,0) steps in them are 0+0+2+2+3=7.


MAPLE

G:=z*(1+z)*(1zz^2)/((13*z+z^2)*(1+z+z^2))^(3/2): Gser:=series(G, z=0, 32): seq(coeff(Gser, z, n), n=0..28);


MATHEMATICA

CoefficientList[Series[x*(1+x)*(1xx^2)/((13*x+x^2)*(1+x+x^2))^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2016 *)


CROSSREFS

Cf. A182884, A182886.
Sequence in context: A244897 A091650 A096240 * A035080 A229188 A091486
Adjacent sequences: A182884 A182885 A182886 * A182888 A182889 A182890


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Dec 11 2010


STATUS

approved



