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
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
KEYWORD
nonn,easy
AUTHOR
Paul Max Payton, Jan 14 2007
EXTENSIONS
Edited and extended by Klaus Brockhaus, Jan 26 2007
STATUS
approved