A127407 Negative value of coefficient of x^(n-2) in the characteristic polynomial of a certain n X n integer circulant matrix.

3, 15, 44, 100, 195, 343, 560, 864, 1275, 1815, 2508, 3380, 4459, 5775, 7360, 9248, 11475, 14079, 17100, 20580, 24563, 29095, 34224, 40000, 46475, 53703, 61740, 70644, 80475, 91295, 103168, 116160, 130339, 145775, 162540, 180708, 200355

Offset: 2

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-2) exists only for n>1, so the sequence starts with a(2). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>1) is multiplied by -1.

Examples

			The circulant matrix for n = 5 is
[1 2 3 4 5]
[5 1 2 3 4]
[4 5 1 2 3]
[3 4 5 1 2]
[2 3 4 5 1]
The characteristic polynomial of this matrix is x^5 - 5*x^4 -100*x^3 - 625*x^2 - 1750*x - 1875. The coefficient of x^(n-2) is -100, hence a(5) = 100.
		

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

• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127408, A127409, A127410, A127411, A127412.

Programs

Magma

[ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-2) : n in [2..38] ]; // Klaus Brockhaus, Jan 27 2007

Magma

[ (n-1) * n^2 * (n+7) / (2 * Factorial(3)) : n in [2..38] ]; // Klaus Brockhaus, Jan 27 2007

Octave

n * (n+1)^2 * (n+8) / (2 * factorial(3)); % Paul Max Payton, Jan 14 2007

PARI

a(n) = {-polcoeff(charpoly(matrix(n,n,i,j,(j-i)%n+1),x),n-2)} \\ Klaus Brockhaus, Jan 27 2007

PARI

a(n) = {(n^4+6*n^3-7*n^2)/(2*3!)} \\ Klaus Brockhaus, Jan 27 2007

Formula

a(n+1) = n*(n+1)^2*(n+8)/(2*3!) for n>=1.

a(n) = ((n-1)^4+10*(n-1)^3+17*(n-1)^2+8*(n-1))/(2*3!) for n>=2.

a(n) = (n^2*(-7+6*n+n^2))/12. G.f.: x^2*(3-x^2)/(1-x)^5. - Colin Barker, May 13 2012

Extensions

Edited by Klaus Brockhaus, Jan 27 2007