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A231188 Coefficient table for the minimal polynomials of 2*sin(2*Pi/n). Rising powers of x.

%I A231188 #10 Dec 18 2013 05:18:45
%S A231188 0,1,0,1,-3,0,1,-2,1,5,0,-5,0,1,-3,0,1,-7,0,14,0,-7,0,1,-2,0,1,-3,0,9,
%T A231188 0,-6,0,1,5,0,-5,0,1,-11,0,55,0,-77,0,44,0,-11,0,1,-1,1,13,0,-91,0,
%U A231188 182,0,-156,0,65,0,-13,0,1,-7,0,14,0,-7,0,1,1,0,-8,0,14,0,-7,0,1,2,0,-4,0,1,17,0,-204,0,714,0,-1122,0,935,0,-442,0,119,0,-17,0,1,-3,0,9,0,-6,0,1
%N A231188 Coefficient table for the minimal polynomials of 2*sin(2*Pi/n). Rising powers of x.
%C A231188 The length of row n is deg(n) + 1 = A093819(n) + 1, that is 2, 2, 3, 2, 5, 3, 7, 3, 7, 5, 11, 2, 13, 7, 9, 5, 17,...
%C A231188 See A181871 for the coefficient table for the integer but non-monic minimal polynomials of sin(2*Pi/n), n>=1, called there pi(n, x). The present minimal polynomials of 2*sin(2*Pi/n) are integer and monic, and they are given by
%C A231188   MP2sin2(n, x) = pi(n, x/2).
%F A231188 a(n,m) = [x^m] MP2sin2(n, x), n>=1, m = 0, 1, ..., A093819(n), with the minimal polynomials of 2*sin(2*Pi/n), given above in a comment in terms of the ones for sin(2*Pi/n).
%e A231188 The table a(n,m) starts:
%e A231188 ---------------------------------------------------------------------------------
%e A231188 n\m   0   1    2  3    4  5     6  7    8  9    10  11   12  13   14 15 16 ...
%e A231188 1:    0   1
%e A231188 2:    0   1
%e A231188 3:   -3   0    1
%e A231188 4:   -2   1
%e A231188 5:    5   0   -5  0    1
%e A231188 6:   -3   0    1
%e A231188 7:   -7   0   14  0   -7  0     1
%e A231188 8:   -2   0    1
%e A231188 9:   -3   0    9  0   -6  0     1
%e A231188 10:   5   0   -5  0    1
%e A231188 11: -11   0   55  0  -77  0    44  0  -11  0     1
%e A231188 12:  -1   1
%e A231188 13:  13   0  -91  0  182  0  -156  0   65  0   -13   0    1
%e A231188 14:  -7   0   14  0   -7  0     1
%e A231188 15:   1   0   -8  0   14  0    -7  0    1
%e A231188 16:   2   0   -4  0    1
%e A231188 17:  17   0 -204  0  714  0 -1122  0  935  0  -442   0  119   0  -17  0  1
%e A231188 ...
%Y A231188 Cf. A093819, A181871, A181872/A181872, A232624.
%K A231188 sign,tabf
%O A231188 1,5
%A A231188 _Wolfdieter Lang_, Nov 29 2013