A127409 Negative value of coefficient of x^(n-4) in the characteristic polynomial of a certain n X n integer circulant matrix.

160, 1750, 10044, 40817, 132608, 367416, 903000, 2020458, 4191264, 8168446, 15107092, 26719875, 45473792, 74834816, 119567664, 186098388, 282948000, 421245846, 615331948, 883458037, 1248597504, 1739375000, 2391126920

Offset: 4

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-4) exists only for n>3, so the sequence starts with a(4). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>3) 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-4) is -1750, hence a(5) = 1750.
		

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 (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127407, A127408, A127410, A127411, A127412.

Programs

Magma

[ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-4) : n in [4..26] ];  // Klaus Brockhaus, Jan 27 2007

Magma

[ (n-3)*(n-2)*(n-1)*n^4*(3*n+13) / (2 * Factorial(5)) : n in [4..26] ]; // Klaus Brockhaus, Jan 27 2007

Octave

n * (n+1) * (n+2) * (n+3)^4 * (3*n + 22) / (2 * factorial(5)); % Paul Max Payton, Jan 14 2007

PARI

a(n) = {-polcoeff(charpoly(matrix(n,n,i,j,(j-i)%n+1),x),n-4)} \\ Klaus Brockhaus, Jan 27 2007

PARI

a(n) = {(3*n^8 - 5*n^7 - 45*n^6 + 125*n^5 - 78*n^4)/(2*5!)} \\ Klaus Brockhaus, Jan 27 2007

Formula

a(n+3) = n*(n+1)*(n+2)*(n+3)^4*(3*n+22)/(2*5!) for n>=1.

a(n) = (3*n^8-5*n^7-45*n^6+125*n^5-78*n^4)/(2*5!) for n>=4.

G.f.: x^4*(160+310*x+54*x^2-19*x^3-x^4)/(1-x)^9. - Colin Barker, May 13 2012

Extensions

Edited, corrected and extended by Klaus Brockhaus, Jan 27 2007