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A284982 Triangle read by rows: coefficients of the Laplacian polynomial of the n-cycle graph C_n.

%I A284982 #11 Feb 16 2025 08:33:43
%S A284982 0,-1,0,-4,1,0,-9,6,-1,0,-16,20,-8,1,0,-25,50,-35,10,-1,0,-36,105,
%T A284982 -112,54,-12,1,0,-49,196,-294,210,-77,14,-1,0,-64,336,-672,660,-352,
%U A284982 104,-16,1,0,-81,540,-1386,1782,-1287,546,-135,18,-1,0,-100,825,-2640,4290,-4004,2275,-800,170,-20,1
%N A284982 Triangle read by rows: coefficients of the Laplacian polynomial of the n-cycle graph C_n.
%C A284982 Extended to n = 1 and n = 2 using the closed form.
%C A284982 Closely related to A127677, which has opposite signs and rows begin with 2 of alternating signs instead of 0.
%H A284982 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>
%H A284982 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaplacianPolynomial.html">Laplacian Polynomial</a>
%e A284982 First few polynomials are
%e A284982 -x
%e A284982 -4*x + x^2
%e A284982 -9*x + 6*x^2 - x^3
%e A284982 which give the triangle of coefficients:
%e A284982 0, -1;
%e A284982 0, -4, 1;
%e A284982 0, -9, 6, -1;
%e A284982 ...
%t A284982 CoefficientList[LucasL[2 Range[10], Sqrt[-x]] - 2, x] // Flatten (* _Eric W. Weisstein_, Apr 06 2017 *)
%t A284982 CoefficientList[2 (ChebyshevT[Range[10], 1 - x/2] - 1), x] // Flatten (* _Eric W. Weisstein_, Apr 06 2017 *)
%t A284982 CoefficientList[FunctionExpand[2 (Cos[Range[10] ArcCos[1 - x/2]] - 1)], x] // Flatten (* _Eric W. Weisstein_, Apr 06 2017 *)
%t A284982 CoefficientList[LinearRecurrence[{3 - x, -3 + x, 1}, {-x, (-4 + x) x, -(-3 + x)^2 x}, 10], x] // Flatten (* _Eric W. Weisstein_, Apr 06 2017 *)
%Y A284982 Cf. A127677 (opposite signs and rows begin with +/2).
%K A284982 sign,easy,tabl
%O A284982 1,4
%A A284982 _Eric W. Weisstein_, Apr 06 2017