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A127407 Negative value of coefficient of x^(n-2) in the characteristic polynomial of a certain n X n integer circulant matrix. 5
3, 15, 44, 100, 195, 343, 560, 864, 1275, 1815, 2508, 3380, 4459, 5775, 7360, 9248, 11475, 14079, 17100, 20580, 24563, 29095, 34224, 40000, 46475, 53703, 61740, 70644, 80475, 91295, 103168, 116160, 130339, 145775, 162540, 180708, 200355 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=2..38.

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

Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127408, A127409, A127410, A127411, A127412.

Sequence in context: A147057 A330322 A101165 * A196237 A177146 A161400

Adjacent sequences:  A127404 A127405 A127406 * A127408 A127409 A127410

KEYWORD

nonn,easy

AUTHOR

Paul Max Payton, Jan 14 2007

EXTENSIONS

Edited by Klaus Brockhaus, Jan 27 2007

STATUS

approved

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Last modified September 17 03:42 EDT 2021. Contains 347478 sequences. (Running on oeis4.)