OFFSET
5,1
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-5) exists only for n>4, so the sequence starts with a(5). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>4) 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
T. D. Noe, Table of n, a(n) for n = 5..1000
FORMULA
a(n+4) = n*(n+1)*(n+2)*(n+3)*(n+4)^5*(4*n+32)/(2*6!) for n>=1.
a(n) = (4*n^10-24*n^9-20*n^8+360*n^7-704*n^6+384*n^5)/(2*6!) for n>=5.
G.f.: x^5*(x^5+53*x^4-82*x^3-2882*x^2-5295*x-1875)/(x-1)^11. [Colin Barker, May 29 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-5) is -1875, hence a(5) = 1875.
PROG
(OCTAVE, MATLAB) n * (n+1) * (n+2) * (n+3) * (n+4)^5 * (4*n + 32) / (2 * factorial(6)); [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-5) : n in [5..24] ] ; 2. [ (n-4)*(n-3)*(n-2)*(n-1)*n^5*(4*n+16) / (2*Factorial(6)) : n in [5..24] ]; // Klaus Brockhaus, Jan 27 2007
(PARI) 1. {for(n=5, 24, print1(-polcoeff(charpoly(matrix(n, n, i, j, (j-i)%n+1), x), n-5), ", "))} 2. {for(n=5, 24, print1((4*n^10-24*n^9-20*n^8+360*n^7-704*n^6+384*n^5)/(2*6!), ", "))} \\ Klaus Brockhaus, Jan 27 2007
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Max Payton, Jan 14 2007
EXTENSIONS
Edited by Klaus Brockhaus, Jan 27 2007
STATUS
approved