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

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A127410 Negative value of coefficient of x^(n-5) 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 (list; graph; refs; listen; history; text; internal format)
 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

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 November 27 14:41 EST 2022. Contains 358405 sequences. (Running on oeis4.)