A346831 Table read by rows, coefficients of the characteristic polynomials of the tangent matrices.

1, 0, 1, -1, 0, 1, 2, -1, -2, 1, 1, 0, -6, 0, 1, 4, 9, -4, -10, 0, 1, -1, 0, 15, 0, -15, 0, 1, 14, -1, -46, 19, 34, -19, -2, 1, 1, 0, -28, 0, 70, 0, -28, 0, 1, 40, 81, -88, -196, 56, 150, -8, -36, 0, 1, -1, 0, 45, 0, -210, 0, 210, 0, -45, 0, 1

Offset

0,7

Comments

The tangent matrix M(n, k) is an N X N matrix defined 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,

for n in [1..N-1] and for k in [0..n-1], and 0 in the main antidiagonal.

The name 'tangent matrix' derives from M(n, k) = signum(tan(Pi*(n + k)/(N + 1))) whenever the right side of this equation is defined.

Links

Table of n, a(n) for n=0..65.

Formula

The rows with even index equal those of A135670.

The determinants of tangent matrices with even dimension are A152011.

Example

Table starts:
[0]  1;
[1]  0,  1;
[2] -1,  0,   1;
[3]  2, -1,  -2,    1;
[4]  1,  0,  -6,    0,   1;
[5]  4,  9,  -4,  -10,   0,   1;
[6] -1,  0,  15,    0, -15,   0,   1;
[7] 14, -1, -46,   19,  34, -19,  -2,   1;
[8]  1,  0, -28,    0,  70,   0, -28,   0, 1;
[9] 40, 81, -88, -196,  56, 150,  -8, -36, 0, 1.
.
The first few tangent matrices:
1       2          3              4                  5
---------------------------------------------------------------
0;   -1  0;    1  -1  0;    1  -1  -1   0;   1   1  -1  -1   0;
      0  1;   -1   0  1;   -1  -1   0   1;   1  -1  -1   0   1;
               0   1  1;   -1   0   1   1;  -1  -1   0   1   1;
                            0   1   1  -1;  -1   0   1   1   1;
                                             0   1   1   1  -1;

Maple

TangentMatrix := 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; M end:
A346831Row := proc(n) if n = 0 then return 1 fi;
   LinearAlgebra:-CharacteristicPolynomial(TangentMatrix(n), x);
   seq(coeff(%, x, k), k = 0..n) end:
seq(A346831Row(n), n = 0..10);

Mathematica

TangentMatrix[N_] := Module[{M, H, n, k},
   M = Array[0&, {N, N}]; H = Quotient[N + 1, 2];
   For[n = 1, n <= N - 1, n++, For[k = 0, k <= n - 1, k++,
      M[[n - k, k + 1]] = If[n < H, 1, -1];
      M[[N - n + k + 1, N - k]] = If[n < N - H, -1, 1]]]; M];
A346831Row[n_] := Module[{c}, If[n == 0,  Return[{1}]];
c = CharacteristicPolynomial[TangentMatrix[n], x];
(-1)^n*CoefficientList[c, x]];
Table[A346831Row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 15 2024, after Peter Luschny *)

Prog

(Julia)
using AbstractAlgebra
function TangentMatrix(N)
    M = zeros(ZZ, N, N)
    H = div(N + 1, 2)
    for n in 1:N - 1
        for k in 0:n - 1
            M[n - k, k + 1] = n < H ? 1 : -1
            M[N - n + k + 1, N - k] = n < N - H ? -1 : 1
        end
    end
M end
function A346831Row(n)
    n == 0 && return [ZZ(1)]
    R, x = PolynomialRing(ZZ, "x")
    S = MatrixSpace(ZZ, n, n)
    M = TangentMatrix(n)
    c = charpoly(R, S(M))
    collect(coefficients(c))
end
for n in 0:9 println(A346831Row(n)) end

Crossrefs

Cf. A135670, A152011, A346837 (generalized tangent matrix).

Sequence in context: A252374 A344569 A341094 * A389848 A161780 A136571

Adjacent sequences: A346828 A346829 A346830 * A346832 A346833 A346834

Keyword

sign,tabl

Author

Peter Luschny, Sep 11 2021

Status

approved