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%I A346837 #15 Apr 15 2024 09:43:53
%S A346837 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,
%T A346837 -14,-17,12,1,-30,-15,0,-1,1,0,28,0,70,0,28,0,1,-40,-63,72,156,40,6,
%U A346837 56,28,0,1
%N A346837 Table read by rows, coefficients of the determinant polynomials of the generalized tangent matrices.
%C A346837 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:
%C A346837 M[n - k, k + 1] = if n < h then 1 otherwise -1,
%C A346837 M[N - n + k + 1, N - k] = if n < N - h then -1 otherwise 1,
%C A346837 and the indeterminate x in the main antidiagonal.
%C A346837 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.
%e A346837 Table starts:
%e A346837 [0] 1;
%e A346837 [1] 0, 1;
%e A346837 [2] -1, 0, -1;
%e A346837 [3] -2, -1, 0, -1;
%e A346837 [4] 1, 0, 6, 0, 1;
%e A346837 [5] -4, 1, 12, 6, 0, 1;
%e A346837 [6] -1, 0, -15, 0, -15, 0, -1;
%e A346837 [7] -14, -17, 12, 1, -30, -15, 0, -1;
%e A346837 [8] 1, 0, 28, 0, 70, 0, 28, 0, 1;
%e A346837 [9] -40, -63, 72, 156, 40, 6, 56, 28, 0, 1.
%e A346837 .
%e A346837 The first few generalized tangent matrices:
%e A346837 1 2 3 4 5
%e A346837 ---------------------------------------------------------------
%e A346837 x; -1 x; 1 -1 x; 1 -1 -1 x; 1 1 -1 -1 x;
%e A346837 x 1; -1 x 1; -1 -1 x 1; 1 -1 -1 x 1;
%e A346837 x 1 1; -1 x 1 1; -1 -1 x 1 1;
%e A346837 x 1 1 -1; -1 x 1 1 1;
%e A346837 x 1 1 1 -1;
%p A346837 GeneralizedTangentMatrix := proc(N) local M, H, n, k;
%p A346837 M := Matrix(N, N); H := iquo(N + 1, 2);
%p A346837 for n from 1 to N - 1 do for k from 0 to n - 1 do
%p A346837 M[n - k, k + 1] := `if`(n < H, 1, -1);
%p A346837 M[N - n + k + 1, N - k] := `if`(n < N - H, -1, 1);
%p A346837 od od; for k from 1 to N do M[k, N-k+1] := x od;
%p A346837 M end:
%p A346837 A346837Row := proc(n) if n = 0 then return 1 fi;
%p A346837 GeneralizedTangentMatrix(n):
%p A346837 LinearAlgebra:-Determinant(%);
%p A346837 seq(coeff(%, x, k), k = 0..n) end:
%p A346837 seq(A346837Row(n), n = 0..9);
%t A346837 GeneralizedTangentMatrix[N_] := Module[{M, H, n, k},
%t A346837 M = Array[0&, {N, N}]; H = Quotient[N + 1, 2];
%t A346837 For[n = 1, n <= N - 1, n++, For[k = 0, k <= n - 1, k++,
%t A346837 M[[n - k, k + 1]] = If[n < H, 1, -1];
%t A346837 M[[N - n + k + 1, N - k]] = If[n < N - H, -1, 1]]];
%t A346837 For[k = 1, k <= N, k++, M[[k, N - k + 1]] = x]; M];
%t A346837 A346837Row[n_] := If[n == 0, {1}, CoefficientList[ Det[
%t A346837 GeneralizedTangentMatrix[n]], x]];
%t A346837 Table[A346837Row[n], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Apr 15 2024, after _Peter Luschny_ *)
%Y A346837 Cf. A011782 (row sums modulo sign), A347596 (alternating row sums), A346831.
%K A346837 sign,tabl
%O A346837 0,7
%A A346837 _Peter Luschny_, Sep 11 2021