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 A187660 Triangle read by rows: T(n,k) = (-1)^(floor(3*k/2))*binomial(floor((n+k)/2),k), 0 <= k <= n. 6
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS 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 LINKS L. E. Jeffery, Danzer matrices Guoce Xin and Yueming Zhong, Proving some conjectures on KekulĂ© numbers for certain benzenoids by using Chebyshev polynomials, arXiv:2201.02376 [math.CO], 2022. FORMULA T(n,k) = (-1)^n*A066170(n,k). abs(T(n,k)) = A046854(n,k) = abs(A066170(n,k)) = abs(A130777(n,k)). abs(T(n,k)) = A065941(n,n-k) = abs(A108299(n,n-k)). EXAMPLE Triangle begins:   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; MAPLE 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 MATHEMATICA t[n_, k_] := (-1)^Floor[3 k/2] Binomial[Floor[(n + k)/2], k]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] (* L. Edson Jeffery, Oct 20 2017 *) CROSSREFS Signed version of A046854. Absolute values of a(n) form a reflected version of A065941, which is considered the main entry. Cf. A046854, A066170, A130777, A267482. Sequence in context: A306209 A267482 A130777 * A066170 A046854 A184957 Adjacent sequences:  A187657 A187658 A187659 * A187661 A187662 A187663 KEYWORD sign,easy,tabl AUTHOR L. Edson Jeffery, Mar 12 2011 EXTENSIONS Edited and corrected by L. Edson Jeffery, Oct 20 2017 STATUS approved

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Last modified July 4 12:21 EDT 2022. Contains 355075 sequences. (Running on oeis4.)