6,1

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-6) exists only for n>5, so the sequence starts with a(6). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>5) is multiplied by -1.

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).

a(n+5) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)^6*(5n+44)/(2 * 7!) for n>=1.

a(n) = (5*n^12 - 56*n^11 + 140*n^10 + 490*n^9 - 2905*n^8 + 4606*n^7 - 2280*n^6)/(2*7!) for n>=6.

The circulant matrix for n = 6 is

[1 2 3 4 5 6]

[6 1 2 3 4 5]

[5 6 1 2 3 4]

[4 5 6 1 2 3]

[3 4 5 6 1 2]

[2 3 4 5 6 1]

The characteristic polynomial of this matrix is x^6 - 6*x^5 -196*x^4 - 1980*x^3 - 10044*x^2 - 25920*x - 27216. The coeffient of x^(n-6) is -27216, hence a(6) = 27216.

(OCTAVE, MATLAB) n * (n+1) * (n+2) * (n+3) * (n+4) * (n+5)^6 * (5*n + 44) / (2*factorial(7)); - Paul M. 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-6) : n in [6..22] ] ; 2. [ (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n^6*(5*n+19) / (2*Factorial(7)) : n in [6..22] ] ; - Klaus Brockhaus, Jan 27 2007

(PARI) 1. {for(n=6, 22, print1(-polcoeff(charpoly(matrix(n, n, i, j, (j-i)%n+1), x), n-6), ", "))} 2. {for(n=6, 22, print1((5*n^12-56*n^11+140*n^10+490*n^9-2905*n^8+4606*n^7-2280*n^6)/(2*7!), ", "))} - Klaus Brockhaus, Jan 27 2007

Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127407, A127408, A127409, A127410, A127412.

Sequence in context: A162144 A159995 A037045 this_sequence A157814 A157820 A032746

Adjacent sequences: A127408 A127409 A127410 this_sequence A127412 A127413 A127414

nonn,easy

Paul M. Payton (paul.payton(AT)lmco.com), Jan 14 2007

Edited by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jan 27 2007