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%I #12 Jun 19 2020 11:24:51
%S 0,1,-2,1,-3,1,2,-4,1,5,-5,1,1,-4,1,-7,14,-7,1,2,-16,20,-8,1,-3,9,-6,
%T 1,1,-12,19,-8,1,-11,55,-77,44,-11,1,1,-16,20,-8,1
%N Irregular triangle read by rows: T(m,k) gives the coefficients of x^k of the minimal polynomials of (2*cos(Pi/(2*m)))^2, for m >= 1.
%C The length of row m is delta(m) + 1 = A055034(m) + 1.
%C For details see A334429, where the formula for the minimal polynomial MPc2(m, x) of 2*cos(Pi/(2*m))^2 = rho(2*m)^2 is given.
%C The companion triangle for odd n is A334432.
%F T(m, k) = [x^k] MPc2even(m, x), with MPc2even(m, x) = Product_{j=1..delta(m)} (x - (2 + R(rpnodd(m)_j, rho(m)))) (evaluated using C(m, rho(m)) = 0), for m >= 2, and MPc2even(1, x) = x. Here R(n, x) is the monic Chebyshev R polynomial with coefficients given in A127672. C(n, x) is the minimal polynomial of rho(n) = 2*cos(Pi/n) given in A187360, and rpnodd(m) is the list of positive odd numbers coprime to m and <= m - 1.
%e The irregular triangle T(m, k) begins:
%e m, n \ k 0 1 2 3 4 5 6 ...
%e -------------------------------------------
%e 1, 2: 0 1
%e 2, 4: -2 1
%e 3, 6: -3 1
%e 4, 8: 2 -4 1
%e 5, 10: 5 -5 1
%e 6, 12: 1 -4 1
%e 7, 14: -7 14 -7 1
%e 8, 16: 2 -16 20 -8 1
%e 9, 18: -3 9 -6 1
%e 10, 20: 1 -12 19 -8 1
%e 11, 22: -11 55 -77 44 -11 1
%e 12, 24: 1 -16 20 -8 1
%e 13, 26: 13 -91 182 -156 65 -13 1
%e 14, 28: 1 -24 86 -104 53 -12 1
%e 15, 30: 1 -8 14 -7 1
%e ...
%Y Cf. A055034, A334429, A334432.
%K sign,tabf,easy
%O 1,3
%A _Wolfdieter Lang_, Jun 15 2020
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