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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

Table of n, a(n) for n=0..77.

L. E. Jeffery, Danzer matrices

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: A225631 A267482 A130777 * A046854 A066170 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 November 21 22:32 EST 2017. Contains 295054 sequences.