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 A127407 #21 Feb 08 2025 21:17:22
%S A127407 3,15,44,100,195,343,560,864,1275,1815,2508,3380,4459,5775,7360,9248,
%T A127407 11475,14079,17100,20580,24563,29095,34224,40000,46475,53703,61740,
%U A127407 70644,80475,91295,103168,116160,130339,145775,162540,180708,200355
%N A127407 Negative value of coefficient of x^(n-2) in the characteristic polynomial of a certain n X n integer circulant matrix.
%C A127407 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 A127407 The coefficient of x^(n-2) exists only for n>1, so the sequence starts with a(2). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>1) is multiplied by -1.
%D A127407 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 A127407 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A127407 a(n+1) = n*(n+1)^2*(n+8)/(2*3!) for n>=1.
%F A127407 a(n) = ((n-1)^4+10*(n-1)^3+17*(n-1)^2+8*(n-1))/(2*3!) for n>=2.
%F A127407 a(n) = (n^2*(-7+6*n+n^2))/12. G.f.: x^2*(3-x^2)/(1-x)^5. - _Colin Barker_, May 13 2012
%e A127407 The circulant matrix for n = 5 is
%e A127407 [1 2 3 4 5]
%e A127407 [5 1 2 3 4]
%e A127407 [4 5 1 2 3]
%e A127407 [3 4 5 1 2]
%e A127407 [2 3 4 5 1]
%e A127407 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-2) is -100, hence a(5) = 100.
%o A127407 (Octave) n * (n+1)^2 * (n+8) / (2 * factorial(3)); % _Paul Max Payton_, Jan 14 2007
%o A127407 (Magma) [ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-2) : n in [2..38] ]; // _Klaus Brockhaus_, Jan 27 2007
%o A127407 (Magma) [ (n-1) * n^2 * (n+7) / (2 * Factorial(3)) : n in [2..38] ]; // _Klaus Brockhaus_, Jan 27 2007
%o A127407 (PARI) a(n) = {-polcoeff(charpoly(matrix(n,n,i,j,(j-i)%n+1),x),n-2)} \\ _Klaus Brockhaus_, Jan 27 2007
%o A127407 (PARI) a(n) = {(n^4+6*n^3-7*n^2)/(2*3!)} \\ _Klaus Brockhaus_, Jan 27 2007
%Y A127407 Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127408, A127409, A127410, A127411, A127412.
%K A127407 nonn,easy
%O A127407 2,1
%A A127407 _Paul Max Payton_, Jan 14 2007
%E A127407 Edited by _Klaus Brockhaus_, Jan 27 2007