A346837 Table read by rows, coefficients of the determinant polynomials of the generalized tangent matrices.

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

Offset: 0

Author

Peter Luschny, Sep 11 2021

Keywords

• sign
• tabl

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.

Examples

			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;
		

Crossrefs

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

Programs

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);

Mathematica

GeneralizedTangentMatrix[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]]];
      For[k = 1, k <= N, k++, M[[k, N - k + 1]] = x]; M];
A346837Row[n_] := If[n == 0, {1}, CoefficientList[ Det[
   GeneralizedTangentMatrix[n]], x]];
Table[A346837Row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Apr 15 2024, after Peter Luschny *)