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Negative value of coefficient of x^(n-6) in the characteristic polynomial of a certain n X n integer circulant matrix.
6

%I #17 Sep 08 2022 08:45:29

%S 27216,453789,3866624,22674816,103500000,393286542,1297410048,

%T 3822832728,10267329072,25518796875,59378761728,130535973152,

%U 273106821312,547049504268,1054272000000,1962916959024,3543150344976,6218839661001,10640820731904,17789062500000

%N Negative value of coefficient of x^(n-6) in the characteristic polynomial of a certain n X n integer circulant matrix.

%C The n X n circulant matrix used here has first row 1 through n and each successive row is a circular rotation of the previous row to the right by one element.

%C The coefficient of x^(n-6) exists only for n>5, so the sequence starts with a(6). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>5) is multiplied by -1.

%D Daniel Zwillinger, ed., "CRC Standard Mathematical Tables and Formulae", 31st Edition, ISBN 1-58488-291, Section 2.6.2.25 (page 141) and Section 2.6.11.3 (page 152).

%H T. D. Noe, <a href="/A127411/b127411.txt">Table of n, a(n) for n = 6..1000</a>

%F a(n+5) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)^6*(5*n+44)/(2*7!) for n>=1.

%F a(n) = (5*n^12 - 56*n^11 + 140*n^10 + 490*n^9 - 2905*n^8 + 4606*n^7 - 2280*n^6)/(2*7!) for n>=6.

%F G.f.: x^6*(x^6 + 131*x^5 + 150*x^4 - 20470*x^3 - 90215*x^2 - 99981*x - 27216)/(x-1)^13. [_Colin Barker_, May 29 2012]

%e The circulant matrix for n = 6 is

%e [1 2 3 4 5 6]

%e [6 1 2 3 4 5]

%e [5 6 1 2 3 4]

%e [4 5 6 1 2 3]

%e [3 4 5 6 1 2]

%e [2 3 4 5 6 1]

%e The characteristic polynomial of this matrix is x^6 - 6*x^5 -196*x^4 - 1980*x^3 - 10044*x^2 - 25920*x - 27216. The coefficient of x^(n-6) is -27216, hence a(6) = 27216.

%o (OCTAVE, MATLAB) n * (n+1) * (n+2) * (n+3) * (n+4) * (n+5)^6 * (5*n + 44) / (2*factorial(7)); [_Paul Max Payton_, Jan 14 2007]

%o (Magma) 1. [ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-6) : n in [6..22] ]; 2. [ (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n^6*(5*n+19) / (2*Factorial(7)) : n in [6..22] ]; // _Klaus Brockhaus_, Jan 27 2007

%o (PARI) 1. {for(n=6,22,print1(-polcoeff(charpoly(matrix(n,n,i,j,(j-i)%n+1),x),n-6),","))} 2. {for(n=6,22,print1((5*n^12-56*n^11+140*n^10+490*n^9-2905*n^8+4606*n^7-2280*n^6)/(2*7!),","))} \\ _Klaus Brockhaus_, Jan 27 2007

%Y Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127407, A127408, A127409, A127410, A127412.

%K nonn,easy

%O 6,1

%A _Paul Max Payton_, Jan 14 2007

%E Edited by _Klaus Brockhaus_, Jan 27 2007