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%I #7 Mar 12 2022 13:16:39
%S 1,1,1,1,3,2,7,3,1,13,10,3,27,29,6,1,61,66,22,4,133,157,75,10,1,287,
%T 398,201,40,5,633,975,538,155,15,1,1407,2334,1506,476,65,6,3121,5631,
%U 4077,1414,280,21,1,6943,13602,10695,4320,966,98,7,15517,32623,27966,12765,3150,462,28,1
%N Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps of weight 2. These are paths 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.
%C Sum of entries in row n is A051286(n).
%C T(n,0)=A098479(n).
%C Sum(k*T(n,k), k=0..n)=A182886(n).
%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(t,z) =1/sqrt(1-2z-2tz^2+z^2+2t*z^3+t^2*z^4-4z^3).
%e T(3,1)=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 have exactly one H step.
%e Triangle starts:
%e 1;
%e 1;
%e 1,1;
%e 3,2;
%e 7,3,1;
%e 13,10,3;
%e 27,29,6,1;
%p G:=1/sqrt(1-2*z-2*t*z^2+z^2+2*t*z^3+t^2*z^4-4*z^3): Gser:=simplify(series(G,z=0,18)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,k),k=0..floor(n/2)) od; # yields sequence in triangular form
%Y Cf. A051286, A098479, A182886.
%K nonn,tabf
%O 0,5
%A _Emeric Deutsch_, Dec 11 2010
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