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.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 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 Table of n, a(n) for n=3..31. Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1). 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 Adjacent sequences: A127405 A127406 A127407 * A127409 A127410 A127411 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.

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