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%I #15 Jul 21 2019 13:25:37
%S 1,0,1,0,1,1,0,0,4,1,0,0,1,9,1,0,0,0,9,16,1,0,0,0,1,36,25,1,0,0,0,0,
%T 16,100,36,1,0,0,0,0,1,100,225,49,1,0,0,0,0,0,25,400,441,64,1,0,0,0,0,
%U 0,1,225,1225,784,81,1,0,0,0,0,0,0,36,1225,3136,1296,100,1,0,0,0,0,0,0,1,441,4900,7056,2025,121,1
%N Triangle read by rows: T(n,k) is the number of lattice paths L_n of weight n having length k (0 <= k <= n). These are paths that start at (0,0) and end on the horizontal axis and whose steps are of the following four kinds: a (1,0)-step with weight 1, a (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1.
%C 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 Sum_{k=0..n} k*T(n,k) = A182879(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 T(n,k) = binomial(n,n-k)^2.
%F G.f. = G(t,z) = ((1-t*z)^2 - 2*t*z^2 - 2*t^2*z^3 + t^2*z^4)^(-1/2).
%e 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 hhh, hH, Hh, ud, and du, having lengths 3, 2, 2, 2, and 2, respectively.
%e Triangle starts:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 0, 4, 1;
%e 0, 0, 1, 9, 1;
%e 0, 0, 0, 9, 16, 1;
%p T:=(n,k)->binomial(k,n-k)^2: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%Y Cf. A051286, A182879.
%K nonn,tabl
%O 0,9
%A _Emeric Deutsch_, Dec 10 2010
%E Keyword tabl added by _Michel Marcus_, Apr 09 2013
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