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 A127409 #19 Feb 08 2025 21:17:26
%S A127409 160,1750,10044,40817,132608,367416,903000,2020458,4191264,8168446,
%T A127409 15107092,26719875,45473792,74834816,119567664,186098388,282948000,
%U A127409 421245846,615331948,883458037,1248597504,1739375000,2391126920
%N A127409 Negative value of coefficient of x^(n-4) in the characteristic polynomial of a certain n X n integer circulant matrix.
%C A127409 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 A127409 The coefficient of x^(n-4) exists only for n>3, so the sequence starts with a(4). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>3) is multiplied by -1.
%D A127409 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 A127409 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A127409 a(n+3) = n*(n+1)*(n+2)*(n+3)^4*(3*n+22)/(2*5!) for n>=1.
%F A127409 a(n) = (3*n^8-5*n^7-45*n^6+125*n^5-78*n^4)/(2*5!) for n>=4.
%F A127409 G.f.: x^4*(160+310*x+54*x^2-19*x^3-x^4)/(1-x)^9. - _Colin Barker_, May 13 2012
%e A127409 The circulant matrix for n = 5 is
%e A127409 [1 2 3 4 5]
%e A127409 [5 1 2 3 4]
%e A127409 [4 5 1 2 3]
%e A127409 [3 4 5 1 2]
%e A127409 [2 3 4 5 1]
%e A127409 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-4) is -1750, hence a(5) = 1750.
%o A127409 (Octave) n * (n+1) * (n+2) * (n+3)^4 * (3*n + 22) / (2 * factorial(5)); % _Paul Max Payton_, Jan 14 2007
%o A127409 (Magma) [ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-4) : n in [4..26] ]; // _Klaus Brockhaus_, Jan 27 2007
%o A127409 (Magma) [ (n-3)*(n-2)*(n-1)*n^4*(3*n+13) / (2 * Factorial(5)) : n in [4..26] ]; // _Klaus Brockhaus_, Jan 27 2007
%o A127409 (PARI) a(n) = {-polcoeff(charpoly(matrix(n,n,i,j,(j-i)%n+1),x),n-4)} \\ _Klaus Brockhaus_, Jan 27 2007
%o A127409 (PARI) a(n) = {(3*n^8 - 5*n^7 - 45*n^6 + 125*n^5 - 78*n^4)/(2*5!)} \\ _Klaus Brockhaus_, Jan 27 2007
%Y A127409 Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127407, A127408, A127410, A127411, A127412.
%K A127409 nonn,easy
%O A127409 4,1
%A A127409 _Paul Max Payton_, Jan 14 2007
%E A127409 Edited, corrected and extended by _Klaus Brockhaus_, Jan 27 2007