This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A127410 #20 Feb 08 2025 21:17:55
%S A127410 1875,25920,184877,917504,3582306,11760000,33820710,87588864,
%T A127410 208295373,461452992,962836875,1908408320,3617795636,6595852032,
%U A127410 11617856508,19845120000,32979115575,53463778368,84747328281,131616866304,200621093750,300598812800,443333396610
%N A127410 Negative value of coefficient of x^(n-5) in the characteristic polynomial of a certain n X n integer circulant matrix.
%C A127410 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.
%C A127410 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.
%D A127410 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).
%H A127410 T. D. Noe, <a href="/A127410/b127410.txt">Table of n, a(n) for n = 5..1000</a>
%F A127410 a(n+4) = n*(n+1)*(n+2)*(n+3)*(n+4)^5*(4*n+32)/(2*6!) for n>=1.
%F A127410 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.
%F A127410 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]
%e A127410 The circulant matrix for n = 5 is
%e A127410 [1 2 3 4 5]
%e A127410 [5 1 2 3 4]
%e A127410 [4 5 1 2 3]
%e A127410 [3 4 5 1 2]
%e A127410 [2 3 4 5 1]
%e A127410 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.
%o A127410 (Octave) n * (n+1) * (n+2) * (n+3) * (n+4)^5 * (4*n + 32) / (2 * factorial(6)); % _Paul Max Payton_, Jan 14 2007
%o A127410 (Magma) [ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-5) : n in [5..24] ]; // _Klaus Brockhaus_, Jan 27 2007
%o A127410 (Magma) [ (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
%o A127410 (PARI) a(n) = {-polcoef(charpoly(matrix(n,n,i,j,(j-i)%n+1),x),n-5)} \\ _Klaus Brockhaus_, Jan 27 2007
%o A127410 (PARI) a(n) = {(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
%Y A127410 Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127407, A127408, A127409, A127411, A127412.
%K A127410 nonn,easy
%O A127410 5,1
%A A127410 _Paul Max Payton_, Jan 14 2007
%E A127410 Edited by _Klaus Brockhaus_, Jan 27 2007