

A127410


Negative value of coefficient of x^(n5) in the characteristic polynomial of a certain n X n integer circulant matrix.


6



1875, 25920, 184877, 917504, 3582306, 11760000, 33820710, 87588864, 208295373, 461452992, 962836875, 1908408320, 3617795636, 6595852032, 11617856508, 19845120000, 32979115575, 53463778368, 84747328281, 131616866304, 200621093750, 300598812800, 443333396610
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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^(n5) 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 158488291, 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^1024*n^920*n^8+360*n^7704*n^6+384*n^5)/(2*6!) for n>=5.
G.f.: x^5*(x^5+53*x^482*x^32882*x^25295*x1875)/(x1)^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^(n5) 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 + (ji) mod n > : i, j in [1..n] ] )), n5) : n in [5..24] ] ; 2. [ (n4)*(n3)*(n2)*(n1)*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, (ji)%n+1), x), n5), ", "))} 2. {for(n=5, 24, print1((4*n^1024*n^920*n^8+360*n^7704*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



