login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A346837 Table read by rows, coefficients of the determinant polynomials of the generalized tangent matrices. 2
1, 0, 1, -1, 0, -1, -2, -1, 0, -1, 1, 0, 6, 0, 1, -4, 1, 12, 6, 0, 1, -1, 0, -15, 0, -15, 0, -1, -14, -17, 12, 1, -30, -15, 0, -1, 1, 0, 28, 0, 70, 0, 28, 0, 1, -40, -63, 72, 156, 40, 6, 56, 28, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

The generalized tangent matrix M(n, k) is an N X N matrix defined for n in [1..N-1] and for k in [0..n-1] with h = floor((N+1)/2) as:

   M[n - k, k + 1] = if n < h then 1 otherwise -1,

   M[N - n + k + 1, N - k] = if n < N - h then -1 otherwise 1,

and the indeterminate x in the main antidiagonal.

The tangent matrix M(n, k) as defined in A346831 is the special case which arises from setting x = 0. The determinant of a generalized tangent matrix M is a polynomial which we call the determinant polynomial of M.

LINKS

Table of n, a(n) for n=0..54.

EXAMPLE

Table starts:

[0]   1;

[1]   0,   1;

[2]  -1,   0,  -1;

[3]  -2,  -1,   0,  -1;

[4]   1,   0,   6,   0,   1;

[5]  -4,   1,  12,   6,   0,   1;

[6]  -1,   0, -15,   0, -15,   0, -1;

[7] -14, -17,  12,   1, -30, -15,  0, -1;

[8]   1,   0,  28,   0,  70,   0, 28,  0, 1;

[9] -40, -63,  72, 156,  40,   6, 56, 28, 0, 1.

.

The first few generalized tangent matrices:

1       2          3              4                  5

---------------------------------------------------------------

x;   -1  x;    1  -1  x;    1  -1  -1   x;   1   1  -1  -1   x;

      x  1;   -1   x  1;   -1  -1   x   1;   1  -1  -1   x   1;

               x   1  1;   -1   x   1   1;  -1  -1   x   1   1;

                            x   1   1  -1;  -1   x   1   1   1;

                                             x   1   1   1  -1;

MAPLE

GeneralizedTangentMatrix := proc(N) local M, H, n, k;

   M := Matrix(N, N); H := iquo(N + 1, 2);

   for n from 1 to N - 1 do for k from 0 to n - 1 do

       M[n - k, k + 1] := `if`(n < H, 1, -1);

       M[N - n + k + 1, N - k] := `if`(n < N - H, -1, 1);

od od; for k from 1 to N do M[k, N-k+1] := x od;

M end:

A346837Row := proc(n) if n = 0 then return 1 fi;

   GeneralizedTangentMatrix(n):

   LinearAlgebra:-Determinant(%);

   seq(coeff(%, x, k), k = 0..n) end:

seq(A346837Row(n), n = 0..9);

CROSSREFS

Cf. A011782 (row sums modulo sign), A347596 (alternating row sums), A346831.

Sequence in context: A233006 A145152 A124327 * A082596 A296139 A321763

Adjacent sequences:  A346834 A346835 A346836 * A346838 A346839 A346840

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Sep 11 2021

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 3 02:46 EST 2021. Contains 349445 sequences. (Running on oeis4.)