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Guoce Xin and Yueming Zhong, <a href="https://arxiv.org/abs/2201.02376">Proving some conjectures on Kekulé numbers for certain benzenoids by using Chebyshev polynomials</a>, arXiv:2201.02376 [math.CO], 2022.

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Triangle read by rows: T(n,k) = (-1)^(floor(3*k/2))*binomial(floor((n+k)/2),k), 0 <= k <= n.

Conjecture: (i) Let n > 1 and N=2*n+1. Row n of T gives the coefficients of the characteristic polynomial p_N(x)=Sum_{k=0..n} T(n,k)*x^(n-k) of the n X n Danzer matrix D_{N,n-1} = {{0,...,0,1}, {0,...,0,1,1}, ..., {0,1,...,1}, {1,...,1}}. (ii) Let S_0(t)=1, S_1(t)=t and S_r(t)=t*S_(r-1)(t)-S_(r-2)(t), r > 1 (cf. A049310). Then p_N(x)=0 has solutions w_{N,j}=S_(n-1)(phi_{N,j}), where phi_{N,j}=2*(-1)^(j+1)*cos(j*Pi/N), j = 1..n. - _L. Edson Jeffery, _, Dec 18 2011

1;

1 , -1;

1 , -1 , -1;

1 , -2 , -1 , 1;

1 , -2 , -3 , 1 , 1;

1 , -3 , -3 , 4 , 1 , -1;

1 , -3 , -6 , 4 , 5 , -1 , -1;

1 , -4 , -6 , 10 , 5 , -6 , -1 , 1;

1 , -4 , -10 , 10 , 15 , -6 , -7 , 1 , 1;

1 , -5 , -10 , 20 , 15 , -21 , -7 , 8 , 1 , -1;

1 , -5 , -15 , 20 , 35 , -21 , -28 , 8 , 9 , -1 , -1;

1 , -6 , -15 , 35 , 35 , -56 , -28 , 36 , 9 , -10 , -1 , 1;

A187660 := proc(n, k): (-1)^(floor(3*k/2))*binomial(floor((n+k)/2), k) end: seq(seq(A187660(n, k), k=0..n), n=0..11); [# _Johannes W. Meijer, _, Aug 08 2011]

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