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 A198637 #15 Feb 16 2025 08:33:16
%S A198637 1,0,1,-4,0,1,-2,-3,0,1,0,0,-4,0,1,-2,5,0,-5,0,1,-4,0,9,0,-6,0,1,-2,
%T A198637 -7,0,14,0,-7,0,1,0,0,-16,0,20,0,-8,0,1,-2,9,0,-30,0,27,0,-9,0,1,-4,0,
%U A198637 25,0,-50,0,35,0,-10,0,1,-2,-11,0,55,0,-77,0,44,0,-11,0,1,0,0,-36,0,105,0,-112,0,54,0,-12,0,1
%N A198637 Coefficient table for the characteristic polynomials of the adjacency matrices of the cycle graphs C_n.
%C A198637 The proof for the row polynomials C(n,x), n>=2, follows by repeated expansion of the determinant, using the Chebyshev S-polynomials recurrence. For n=0 one defines C(0,x):=1, and for n=1 one has C(1,x)=x.
%C A198637 Modulo signs and first terms, essentially the same as A123343. - _Eric W. Weisstein_, Apr 05 2017
%H A198637 Eric Weisstein's Mathworld: <a href="https://mathworld.wolfram.com/AdjacencyMatrix.html">Adjacency Matrix</a>
%H A198637 Eric Weisstein's Mathworld: <a href="https://mathworld.wolfram.com/CharacteristicPolynomial.html">Characteristic Polynomial</a>
%H A198637 Eric Weisstein's Mathworld: <a href="https://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>
%F A198637 a(n,m)=[x^m]C(n,x), with C(0,x):=1, C(1,x)=x, and
%F A198637 C(n,x) = 2*(T(n,x/2)-1) = R(n,x)-2 , for n>=2, with Chebyshev's T-polynomial or its monic integer version R(n,x) (usually called Chebyshev C-polynomials) with coefficient table A127672, from which a formula for a(n,m) follows. Only the column m=0 differs.
%e A198637 The table begins
%e A198637 n\m 0 1 2 3 4 5 6 7 8 9 10 ...
%e A198637 0: 1
%e A198637 1: 0 1
%e A198637 2: -4 0 1
%e A198637 3: -2 -3 0 1
%e A198637 4: 0 0 -4 0 1
%e A198637 5: -2 5 0 -5 0 1
%e A198637 6: -4 0 9 0 -6 0 1
%e A198637 7: -2 -7 0 14 0 -7 0 1
%e A198637 8: 0 0 -16 0 20 0 -8 0 1
%e A198637 9: -2 9 0 -30 0 27 0 -9 0 1
%e A198637 10: -4 0 25 0 -50 0 35 0 -10 0 1
%e A198637 ...
%e A198637 C(4,x) = -4*x^2 - x^4, with zeros 2, 0, -2, 0.
%e A198637 C(5,x) =-2 + 5*x - 5*x^3 + x^5, with zeros 2, phi-1, -phi, -phi and phi-1, with the golden section phi:=(1+sqrt(5))/2.
%e A198637 The adjacency matrix for C_1 is [[0]],
%e A198637 for C_2 it is [[0,2],[2,0]], and for C_3 it is [[0,1,1],[1,0,1],[1,1,0]].
%t A198637 Flatten[{{1}, {0, 1}, Table[(-1)^n CoefficientList[CharacteristicPolynomial[AdjacencyMatrix[CycleGraph[n]], x], x], {n, 2, 10}]}] (* _Eric W. Weisstein_, Apr 05 2017 *)
%t A198637 Flatten[{{1}, {0, 1}, Table[CoefficientList[2 (ChebyshevT[n, x/2] - 1), x], {n, 2, 10}]}] (* _Eric W. Weisstein_, Apr 05 2017 *)
%Y A198637 Cf. A127672.
%Y A198637 Cf. A123343 (essentially the same sequence).
%K A198637 sign,easy,tabl
%O A198637 0,4
%A A198637 _Wolfdieter Lang_, Nov 08 2011