A127408 Negative value of coefficient of x^(n-3) in the characteristic polynomial of a certain n X n integer circulant matrix.

18, 144, 625, 1980, 5145, 11648, 23814, 45000, 79860, 134640, 217503, 338884, 511875, 752640, 1080860, 1520208, 2098854, 2850000, 3812445, 5031180, 6558013, 8452224, 10781250, 13621400, 17058600, 21189168, 26120619, 31972500, 38877255

Offset

3,1

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-3) exists only for n>2, so the sequence starts with a(3). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>2) is multiplied by -1.

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

Table of n, a(n) for n=3..31.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(n+2) = n*(n+1)*(n+2)^3*(2n+14)/(2 * 4!) for n>=1.

a(n) = (n^6+2*n^5-13*n^4+10*n^3)/4! for n>=3.

G.f.: x^3*(3-x)*(6+8*x+x^2)/(1-x)^7. [Colin Barker, May 13 2012]

Example

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-3) is -625, hence a(5) = 625.

Prog

(OCTAVE, MATLAB) n * (n+1) * (n+2)^3 * (2*n + 14) / (2 * factorial(4)); [Paul Max Payton, Jan 14 2007]
(Magma) 1. [ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-3) : n in [3..31] ] ; 2. [ (n-2) * (n-1) * n^3 * (2*(n-2) + 14) / (2 * Factorial(4)) : n in [3..31] ] ; // Klaus Brockhaus, Jan 26 2007
(PARI) 1. {for(n=3, 31, print1(-polcoeff(charpoly(matrix(n, n, i, j, (j-i)%n+1), x), n-3), ", "))} 2. {for(n=3, 31, print1((n^6+2*n^5-13*n^4+10*n^3)/4!, ", "))} \\ Klaus Brockhaus, Jan 26 2007

Crossrefs

Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127407, A127409, A127410, A127411, A127412.

Sequence in context: A126513 A232820 A143992 * A008452 A126900 A178759

Adjacent sequences: A127405 A127406 A127407 * A127409 A127410 A127411

Keyword

nonn,easy

Author

Paul Max Payton, Jan 14 2007

Extensions

Edited and extended by Klaus Brockhaus, Jan 26 2007

Status

approved