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%I #21 Mar 27 2017 03:53:07
%S 0,0,0,2,6,18,56,162,462,1306,3648,10116,27892,76524,209112,569506,
%T 1546542,4189314,11323480,30548190,82272330,221240070,594131160,
%U 1593553452,4269391596,11426761548,30554523096,81631135502,217918012002
%N Number of (1,1)-steps in all weighted lattice paths in L_n.
%C L_n is the set of lattice 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; 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="/A182881/b182881.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) = Sum_{k>=0} k*A182880(n,k).
%F G.f.: 2*z^3/[(1-3*z+z^2)*(1+z+z^2)]^(3/2).
%F a(n) ~ ((3 + sqrt(5))/2)^n * sqrt(n) / (2*sqrt(Pi)*5^(3/4)). - _Vaclav Kotesovec_, Mar 06 2016
%F Conjecture: (-n+3)*a(n) +(2*n-5)*a(n-1) +(n-2)*a(n-2) +(2*n-3)*a(n-3) +(-n+1)*a(n-4)=0. - _R. J. Mathar_, Jun 14 2016
%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, containing a total of 1+1+0+0+0=2 u steps.
%p g:=2*z^3/((1-3*z+z^2)*(1+z+z^2))^(3/2): gser:=series(g,z=0,32): seq(coeff(gser,z,n),n=0..28);
%t CoefficientList[Series[2*x^3/((1-3*x+x^2)*(1+x+x^2))^(3/2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 06 2016 *)
%o (PARI) z='z+O('z^50); concat([0,0,0], Vec(2*z^3/((1-3*z+z^2)*(1+z+z^2))^(3/2))) \\ _G. C. Greubel_, Mar 25 2017
%Y Cf. A182880.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Dec 11 2010
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