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A127408 Negative value of coefficient of x^(n-3) in the characteristic polynomial of a certain n X n integer circulant matrix.

%I A127408 #23 Feb 08 2025 21:17:44
%S A127408 18,144,625,1980,5145,11648,23814,45000,79860,134640,217503,338884,
%T A127408 511875,752640,1080860,1520208,2098854,2850000,3812445,5031180,
%U A127408 6558013,8452224,10781250,13621400,17058600,21189168,26120619,31972500,38877255
%N A127408 Negative value of coefficient of x^(n-3) in the characteristic polynomial of a certain n X n integer circulant matrix.
%C A127408 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 A127408 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.
%D A127408 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 A127408 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A127408 a(n+2) = n*(n+1)*(n+2)^3*(2n+14)/(2 * 4!) for n>=1.
%F A127408 a(n) = (n^6+2*n^5-13*n^4+10*n^3)/4! for n>=3.
%F A127408 G.f.: x^3*(3-x)*(6+8*x+x^2)/(1-x)^7. - _Colin Barker_, May 13 2012
%e A127408 The circulant matrix for n = 5 is
%e A127408 [1 2 3 4 5]
%e A127408 [5 1 2 3 4]
%e A127408 [4 5 1 2 3]
%e A127408 [3 4 5 1 2]
%e A127408 [2 3 4 5 1]
%e A127408 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.
%o A127408 (Octave) n * (n+1) * (n+2)^3 * (2*n + 14) / (2 * factorial(4)); % _Paul Max Payton_, Jan 14 2007
%o A127408 (Magma) [ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-3) : n in [3..31] ] ; // _Klaus Brockhaus_, Jan 26 2007
%o A127408 (Magma) [ (n-2) * (n-1) * n^3 * (2*(n-2) + 14) / (2 * Factorial(4)) : n in [3..31] ] ; // _Klaus Brockhaus_, Jan 26 2007
%o A127408 (PARI) a(n) = {-polcoef(charpoly(matrix(n,n,i,j,(j-i)%n+1),x),n-3)} \\ _Klaus Brockhaus_, Jan 26 2007
%o A127408 (PARI) a(n) = {(n^6+2*n^5-13*n^4+10*n^3)/4!} \\ _Klaus Brockhaus_, Jan 26 2007
%Y A127408 Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127407, A127409, A127410, A127411, A127412.
%K A127408 nonn,easy
%O A127408 3,1
%A A127408 _Paul Max Payton_, Jan 14 2007
%E A127408 Edited and extended by _Klaus Brockhaus_, Jan 26 2007