A127411 Negative value of coefficient of x^(n-6) in the characteristic polynomial of a certain n X n integer circulant matrix.

27216, 453789, 3866624, 22674816, 103500000, 393286542, 1297410048, 3822832728, 10267329072, 25518796875, 59378761728, 130535973152, 273106821312, 547049504268, 1054272000000, 1962916959024, 3543150344976, 6218839661001, 10640820731904, 17789062500000

Offset: 6

Author

Paul Max Payton, Jan 14 2007

Keywords

• nonn
• easy

Comments

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.

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.

Examples

			The circulant matrix for n = 6 is
[1 2 3 4 5 6]
[6 1 2 3 4 5]
[5 6 1 2 3 4]
[4 5 6 1 2 3]
[3 4 5 6 1 2]
[2 3 4 5 6 1]
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.
		

References

• 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).

Links

• T. D. Noe, Table of n, a(n) for n = 6..1000

Crossrefs

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

Programs

Magma

[ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-6) : n in [6..22] ]; // Klaus Brockhaus, Jan 27 2007

Magma

[ (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

Octave

n * (n+1) * (n+2) * (n+3) * (n+4) * (n+5)^6 * (5*n + 44) / (2*factorial(7)); % Paul Max Payton, Jan 14 2007

PARI

a(n) = {-polcoef(charpoly(matrix(n,n,i,j,(j-i)%n+1),x),n-6)} \\ Klaus Brockhaus, Jan 27 2007

PARI

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!)} \\ Klaus Brockhaus, Jan 27 2007

Formula

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

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.

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

Extensions

Edited by Klaus Brockhaus, Jan 27 2007