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%I A346831 #27 Apr 15 2024 09:44:47
%S A346831 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,
%T A346831 14,-1,-46,19,34,-19,-2,1,1,0,-28,0,70,0,-28,0,1,40,81,-88,-196,56,
%U A346831 150,-8,-36,0,1,-1,0,45,0,-210,0,210,0,-45,0,1
%N A346831 Table read by rows, coefficients of the characteristic polynomials of the tangent matrices.
%C A346831 The tangent matrix M(n, k) is an N X N matrix defined with h = floor((N+1)/2) as:
%C A346831 M[n - k, k + 1] = if n < h then 1 otherwise -1,
%C A346831 M[N - n + k + 1, N - k] = if n < N - h then -1 otherwise 1,
%C A346831 for n in [1..N-1] and for k in [0..n-1], and 0 in the main antidiagonal.
%C A346831 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.
%F A346831 The rows with even index equal those of A135670.
%F A346831 The determinants of tangent matrices with even dimension are A152011.
%e A346831 Table starts:
%e A346831 [0] 1;
%e A346831 [1] 0, 1;
%e A346831 [2] -1, 0, 1;
%e A346831 [3] 2, -1, -2, 1;
%e A346831 [4] 1, 0, -6, 0, 1;
%e A346831 [5] 4, 9, -4, -10, 0, 1;
%e A346831 [6] -1, 0, 15, 0, -15, 0, 1;
%e A346831 [7] 14, -1, -46, 19, 34, -19, -2, 1;
%e A346831 [8] 1, 0, -28, 0, 70, 0, -28, 0, 1;
%e A346831 [9] 40, 81, -88, -196, 56, 150, -8, -36, 0, 1.
%e A346831 .
%e A346831 The first few tangent matrices:
%e A346831 1 2 3 4 5
%e A346831 ---------------------------------------------------------------
%e A346831 0; -1 0; 1 -1 0; 1 -1 -1 0; 1 1 -1 -1 0;
%e A346831 0 1; -1 0 1; -1 -1 0 1; 1 -1 -1 0 1;
%e A346831 0 1 1; -1 0 1 1; -1 -1 0 1 1;
%e A346831 0 1 1 -1; -1 0 1 1 1;
%e A346831 0 1 1 1 -1;
%p A346831 TangentMatrix := proc(N) local M, H, n, k;
%p A346831 M := Matrix(N, N); H := iquo(N + 1, 2);
%p A346831 for n from 1 to N - 1 do for k from 0 to n - 1 do
%p A346831 M[n - k, k + 1] := `if`(n < H, 1, -1);
%p A346831 M[N - n + k + 1, N - k] := `if`(n < N - H, -1, 1);
%p A346831 od od; M end:
%p A346831 A346831Row := proc(n) if n = 0 then return 1 fi;
%p A346831 LinearAlgebra:-CharacteristicPolynomial(TangentMatrix(n), x);
%p A346831 seq(coeff(%, x, k), k = 0..n) end:
%p A346831 seq(A346831Row(n), n = 0..10);
%t A346831 TangentMatrix[N_] := Module[{M, H, n, k},
%t A346831 M = Array[0&, {N, N}]; H = Quotient[N + 1, 2];
%t A346831 For[n = 1, n <= N - 1, n++, For[k = 0, k <= n - 1, k++,
%t A346831 M[[n - k, k + 1]] = If[n < H, 1, -1];
%t A346831 M[[N - n + k + 1, N - k]] = If[n < N - H, -1, 1]]]; M];
%t A346831 A346831Row[n_] := Module[{c}, If[n == 0, Return[{1}]];
%t A346831 c = CharacteristicPolynomial[TangentMatrix[n], x];
%t A346831 (-1)^n*CoefficientList[c, x]];
%t A346831 Table[A346831Row[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Apr 15 2024, after _Peter Luschny_ *)
%o A346831 (Julia)
%o A346831 using AbstractAlgebra
%o A346831 function TangentMatrix(N)
%o A346831 M = zeros(ZZ, N, N)
%o A346831 H = div(N + 1, 2)
%o A346831 for n in 1:N - 1
%o A346831 for k in 0:n - 1
%o A346831 M[n - k, k + 1] = n < H ? 1 : -1
%o A346831 M[N - n + k + 1, N - k] = n < N - H ? -1 : 1
%o A346831 end
%o A346831 end
%o A346831 M end
%o A346831 function A346831Row(n)
%o A346831 n == 0 && return [ZZ(1)]
%o A346831 R, x = PolynomialRing(ZZ, "x")
%o A346831 S = MatrixSpace(ZZ, n, n)
%o A346831 M = TangentMatrix(n)
%o A346831 c = charpoly(R, S(M))
%o A346831 collect(coefficients(c))
%o A346831 end
%o A346831 for n in 0:9 println(A346831Row(n)) end
%Y A346831 Cf. A135670, A152011, A346837 (generalized tangent matrix).
%K A346831 sign,tabl
%O A346831 0,7
%A A346831 _Peter Luschny_, Sep 11 2021