login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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: 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 16:12 EST 2019. Contains 329753 sequences. (Running on oeis4.)