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.
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);
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 *)
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Sep 11 2021
STATUS
approved