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Triangle read by rows: characteristic polynomials of Z/nZ addition tables considered as matrices.
0

%I #19 Dec 23 2018 14:45:17

%S 1,1,0,1,0,-1,1,-3,-3,9,1,-4,-20,32,96,1,-10,-25,250,125,-1250,1,-12,

%T -93,576,2592,-5184,-19440,1,-21,-98,0,2058,2401,50421,16807,352947,1,

%U -24,-272,3840,24064,-147456,-753664,1572864,7340032

%N Triangle read by rows: characteristic polynomials of Z/nZ addition tables considered as matrices.

%C The n-th row of the triangle is the characteristic polynomial of n-th order, Z/nZ addition table considered as a matrix.

%C The determinant of n-th order matrix (irrespective of sign) is the rightmost term of n-th order polynomial.

%e Addition table Z/3Z = [0,1,2; 1,2,0; 2,0,1]. Considered as a matrix, the characteristic polynomial = x^3 - 3x^2 - 3x + 9.

%e First few characteristic polynomials are:

%e 1;

%e x + 0;

%e x^2 + 0x - 1;

%e x^3 - 3x^2 - 3x + 9;

%e x^4 - 4x^3 - 20x^2 + 32x + 96;

%e x^5 - 10x^4 - 25x^3 + 250x^2 + 125x - 1250;

%e x^6 - 12x^5 - 93x^4 + 576x^3 + 2592x^2 - 5194x - 19440;

%e x^7 - 21x^6 - 98x^5 + 0x^4 + 2058x^3 - 50421x^2 - 16807x + 352947;

%e x^8 - 24x^7 - 272x^6 + 3840x^5 + 24064x^4 - 147456x^3 - 753664x^2 + 1572864x + 7340032;

%Y Cf. A007070, A140044, A140045, A095897.

%K sign,tabl

%O 0,8

%A _Gary W. Adamson_, May 03 2008

%E Missing (zero) terms inserted by _Michel Marcus_, Apr 14 2013