OFFSET
2,1
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-2) exists only for n>1, so the sequence starts with a(2). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>1) 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 (5,-10,10,-5,1).
FORMULA
a(n+1) = n*(n+1)^2*(n+8)/(2*3!) for n>=1.
a(n) = ((n-1)^4+10*(n-1)^3+17*(n-1)^2+8*(n-1))/(2*3!) for n>=2.
a(n) = (n^2*(-7+6*n+n^2))/12. G.f.: x^2*(3-x^2)/(1-x)^5. [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-2) is -100, hence a(5) = 100.
PROG
(OCTAVE, MATLAB) n * (n+1)^2 * (n+8) / (2 * factorial(3)); [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-2) : n in [2..38] ] ; 2. [ (n-1) * n^2 * (n+7) / (2 * Factorial(3)) : n in [2..38] ]; // Klaus Brockhaus, Jan 27 2007
(PARI) 1. {for(n=2, 38, print1(-polcoeff(charpoly(matrix(n, n, i, j, (j-i)%n+1), x), n-2), ", "))} 2. {for(n=2, 38, print1((n^4+6*n^3-7*n^2)/(2*3!), ", "))} \\ 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