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%I #14 Jul 22 2022 12:07:14
%S 1,0,1,2,3,8,17,38,89,206,485,1152,2751,6614,15983,38798,94569,231342,
%T 567771,1397562,3449285,8533886,21161001,52579900,130896887,326440746,
%U 815437967,2040049514,5111051473,12822135138,32207384995,80995950182,203917464635
%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)=A182888(n,0).
%D M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
%D 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.
%F G.f.: G(z) =1/( z+sqrt((1+z+z^2)(1-3z+z^2)) ).
%F a(n) ~ sqrt(7*sqrt(5)-15) * ((3 + sqrt(5))/2)^(n+2) / (sqrt(2*Pi) * n^(3/2)). - _Vaclav Kotesovec_, Mar 06 2016
%F Equivalently, a(n) ~ 5^(1/4) * phi^(2*n + 2) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 06 2021
%F D-finite with recurrence n*a(n) +(-4*n+3)*a(n-1) +(n-3)*a(n-2) +(2*n-3)*a(n-3) +12*(n-3)*a(n-4) +(2*n-9)*a(n-5) +(n-3)*a(n-6) +(-4*n+21)*a(n-7) +(n-6)*a(n-8)=0. - _R. J. Mathar_, Jul 22 2022
%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 h steps at level 0.
%p 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);
%t CoefficientList[Series[1/(x+Sqrt[(1+x+x^2)(1-3x+x^2)]),{x,0,40}],x] (* _Harvey P. Dale_, Jun 16 2013 *)
%Y Cf. A182888.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Dec 11 2010
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