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A187660 Triangle in which T(n,k) = (-1)^(floor[3*k/2])*binomial[floor[(n+k)/2],k] is entry k of row n, where 0<=k<=n. 7
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, 1, -6, -21, 35, 70, -56 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

(Start) Let T(n,k) denote entry k (0<=k<=n) in row n of the triangle which begins

{1}

{1,-1}

{1,-1,-1}

{1,-2,-1,1}

{1,-2,-3,1,1},...

Let n>1 and N=2*n+1. Row n gives the coefficients of the characteristic polynomial p_N(x)=Sum[k=0..n, T(n,k)*x^{(n-1)*(n-k)}] of the n X n Danzer matrix (see [Jeffery])

D_{n-1} = [0,...,0,1; 0,...,0,1,1; ...; 0,1,...,1; 1,...,1].

Let {S_r(t)} be a sequence of polynomials with recurrence relation

S_0(t)=1, S_1(t)=t, S_r(t)=t*S_(r-1)(t)-S_(r-2)(t) (r>1).

(cf. A049310.) Then p_N(x)=0 has solutions w_j=S_(n-1)(phi_j), where phi_j=2*cos((2*j-1)*Pi/N), j = 1,2,...,n; that is, the w_j are the eigenvalues of D_{n-1}. (See the example below.) (End)

Also taking the above exponents in reverse order gives the characteristic polynomial Sum[k=0..n, T(n,k)*x^{(n-1)*k}] for the n X n Danzer matrix D_{1} (again see [Jeffery]). - L. Edson Jeffery, Dec 18 2011

LINKS

Table of n, a(n) for n=0..83.

L. E. Jeffery, Danzer matrices

FORMULA

T(n,k) = (-1)^(floor(3*k/2))*binomial(floor((n+k)/2),k)

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

For n=3 (N=7), k = 0,1,2,3, {T(3,k)} = {1,-2,-1,1}, giving the coefficients of the characteristic function p_7(x)=x^6-2*x^4-x^2+1=0 for the 3 X 3 Danzer matrix D_2=[0,0,1;0,1,1;1,1,1] with eigenvalues

w_j=S_2(phi_j)=[2*cos((2*j-1)*Pi/7)]^2-1, j=1,2,3.

Hence entries in row n>0 of the triangle are also given, up to a sign, by the elementary symmetric polynomials e_k in the w_j: defining e_0=1, for all n, then, again for n=3 (N=7), we have

e_0=T(3,0)=1,

e_1=-T(3,1)=w_1+w_2+w_3=2,

e_2=T(3,2)=w_1*w_2+w_1*w_3+w_2*w_3=-1,

e_3=-T(3,3)=w_1*w_2*w_3=-1,

so also p_7(x)=Sum[k=0..3, (-1)^k*e_k*x^{2*(3-k)}].

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]

CROSSREFS

Signed version of A046854.

Absolute values of a(n) form a reflected version of A065941, which is considered the main entry.

Cf. A066170, A130777.

Sequence in context: A267482 A130777 A046854 * A066170 A184957 A228349

Adjacent sequences:  A187657 A187658 A187659 * A187661 A187662 A187663

KEYWORD

sign,easy,tabl

AUTHOR

L. Edson Jeffery, Mar 12 2011

EXTENSIONS

Edited by L. Edson Jeffery, Jan 04 2013

STATUS

approved

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Last modified May 25 19:28 EDT 2017. Contains 287059 sequences.