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 A346831 Table read by rows, coefficients of the characteristic polynomials of the tangent matrices. 7
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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); 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 * A161780 A136571 A178562 Adjacent sequences:  A346828 A346829 A346830 * A346832 A346833 A346834 KEYWORD sign,tabl AUTHOR Peter Luschny, Sep 11 2021 STATUS approved

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Last modified May 23 04:06 EDT 2022. Contains 353959 sequences. (Running on oeis4.)