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%I #20 Dec 06 2021 09:01:41
%S 1,1,1,3,7,15,35,83,197,473,1145,2787,6819,16759,41345,102341,254075,
%T 632437,1577967,3945517,9884379,24806201,62355121,156974319,395712759,
%U 998809135,2524043569,6385400005,16170553755,40990092629,103997889735
%N Number of weighted lattice paths in L_n having no (1,0)-steps of weight 2 at level 0.
%C 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.
%H G. C. Greubel, <a href="/A182892/b182892.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Bona and A. Knopfmacher, <a href="http://dx.doi.org/10.1007/s00026-010-0060-7">On the probability that certain compositions have the same number of parts</a>, Ann. Comb., 14 (2010), 291-306.
%H E. Munarini, N. Zagaglia Salvi, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00378-3">On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns</a>, Discrete Mathematics 259 (2002), 163-177.
%F a(n) = A182891(n,0).
%F G.f.: G(z) =1/( z^2+sqrt((1+z+z^2)*(1-3*z+z^2)) ).
%F a(n) ~ sqrt(360 + 161*sqrt(5)) * ((3 + sqrt(5))/2)^n / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 06 2016. Equivalently, a(n) ~ 5^(1/4) * phi^(2*n + 6) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 06 2021
%F Conjecture: n*a(n) +(n-2)*a(n-1) +2*(-9*n+16)*a(n-2) +5*(2*n-5)*a(n-3) +(10*n-33) *a(n-4) +2*(26*n-109)*a(n-5) +(13*n-37)*a(n-6) +(13*n-63) *a(n-7) +10*(-n+7) *a(n-8)=0. - _R. J. Mathar_, Jun 14 2016
%e a(3)=3. 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; three of them, namely ud, du, and hhh, have no H-steps at level 0.
%p G:=1/(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..30);
%t CoefficientList[Series[1/(x^2+Sqrt[(1+x+x^2)(1-3x+x^2)]),{x,0,30}],x] (* _Harvey P. Dale_, Aug 25 2012 *)
%o (PARI) z='z+O('z^50); Vec(1/( z^2+sqrt((1+z+z^2)*(1-3*z+z^2)) )) \\ _G. C. Greubel_, Mar 26 2017
%Y Cf. A182891.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Dec 12 2010
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