

A127408


Negative value of coefficient of x^(n3) in the characteristic polynomial of a certain n X n integer circulant matrix.


5



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
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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^(n3) 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 158488291, Section 2.6.2.25 (page 141) and Section 2.6.11.3 (page 152).


LINKS



FORMULA

a(n+2) = n*(n+1)*(n+2)^3*(2n+14)/(2 * 4!) for n>=1.
a(n) = (n^6+2*n^513*n^4+10*n^3)/4! for n>=3.
G.f.: x^3*(3x)*(6+8*x+x^2)/(1x)^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^(n3) 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 + (ji) mod n > : i, j in [1..n] ] )), n3) : n in [3..31] ] ; 2. [ (n2) * (n1) * n^3 * (2*(n2) + 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, (ji)%n+1), x), n3), ", "))} 2. {for(n=3, 31, print1((n^6+2*n^513*n^4+10*n^3)/4!, ", "))} \\ Klaus Brockhaus, Jan 26 2007


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



