The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A127408 Negative value of coefficient of x^(n-3) in the characteristic polynomial of a certain n X n integer circulant matrix. 5
18, 144, 625, 1980, 5145, 11648, 23814, 45000, 79860, 134640, 217503, 338884, 511875, 752640, 1080860, 1520208, 2098854, 2850000, 3812445, 5031180, 6558013, 8452224, 10781250, 13621400, 17058600, 21189168, 26120619, 31972500, 38877255 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,1
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-3) exists only for n>2, so the sequence starts with a(3). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>2) 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
FORMULA
a(n+2) = n*(n+1)*(n+2)^3*(2n+14)/(2 * 4!) for n>=1.
a(n) = (n^6+2*n^5-13*n^4+10*n^3)/4! for n>=3.
G.f.: x^3*(3-x)*(6+8*x+x^2)/(1-x)^7. [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-3) is -625, hence a(5) = 625.
PROG
(OCTAVE, MATLAB) n * (n+1) * (n+2)^3 * (2*n + 14) / (2 * factorial(4)); [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-3) : n in [3..31] ] ; 2. [ (n-2) * (n-1) * n^3 * (2*(n-2) + 14) / (2 * Factorial(4)) : n in [3..31] ] ; // Klaus Brockhaus, Jan 26 2007
(PARI) 1. {for(n=3, 31, print1(-polcoeff(charpoly(matrix(n, n, i, j, (j-i)%n+1), x), n-3), ", "))} 2. {for(n=3, 31, print1((n^6+2*n^5-13*n^4+10*n^3)/4!, ", "))} \\ Klaus Brockhaus, Jan 26 2007
CROSSREFS
Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127407, A127409, A127410, A127411, A127412.
Sequence in context: A126513 A232820 A143992 * A008452 A126900 A178759
KEYWORD
nonn,easy
AUTHOR
Paul Max Payton, Jan 14 2007
EXTENSIONS
Edited and extended by Klaus Brockhaus, Jan 26 2007
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 19 00:35 EDT 2024. Contains 372666 sequences. (Running on oeis4.)