A127410 Negative value of coefficient of x^(n-5) in the characteristic polynomial of a certain n X n integer circulant matrix.

1875, 25920, 184877, 917504, 3582306, 11760000, 33820710, 87588864, 208295373, 461452992, 962836875, 1908408320, 3617795636, 6595852032, 11617856508, 19845120000, 32979115575, 53463778368, 84747328281, 131616866304, 200621093750, 300598812800, 443333396610

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

Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127407, A127408, A127409, A127411, A127412.

Sequence in context: A139668 A244019 A054818 * A237570 A045201 A020407

Adjacent sequences: A127407 A127408 A127409 * A127411 A127412 A127413

Keyword

nonn,easy

Author

Paul Max Payton, Jan 14 2007

Extensions

Edited by Klaus Brockhaus, Jan 27 2007

Status

approved