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.
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 12 2010
STATUS
approved