%I
%S 1,0,0,2,2,4,12,24,54,130,300,706,1686,4028,9686,23426,56866,138584,
%T 338940,831508,2045736,5046240,12477290,30919122,76774382,190995224,
%U 475979602,1188125394,2970282794,7436232760,18641883396,46792219972,117590713254
%N 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.
%C a(n)=A182893(n,0).
%D M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291306.
%D E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163177.
%F G.f.: G(z) =1/( z+z^2+sqrt((1+z+z^2)(13z+z^2)) ).
%F a(n) ~ sqrt(105 + 47*sqrt(5)) * ((3 + sqrt(5))/2)^n / (5*sqrt(2*Pi)*n^(3/2)).  _Vaclav Kotesovec_, Mar 06 2016
%F Conjecture: n*a(n) +(4*n+3)*a(n1) +(n3)*a(n2) 3*a(n3) +3*(5*n14)*a(n4) +6*(n3)*a(n5) +6*(n4)*a(n6) +4*(n+6)*a(n7)=0.  _R. J. Mathar_, Jun 14 2016
%F 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
%e 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.
%p G:=1/(z+z^2+sqrt((1+z+z^2)*(13*z+z^2))): Gser:=series(G,z=0,35): seq(coeff(Gser,z,n),n=0..32);
%t CoefficientList[Series[1/(x+x^2+Sqrt[(1+x+x^2)*(13*x+x^2)]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 06 2016 *)
%Y Cf. A182893.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Dec 12 2010
