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

0, -1, 0, -4, 1, 0, -9, 6, -1, 0, -16, 20, -8, 1, 0, -25, 50, -35, 10, -1, 0, -36, 105, -112, 54, -12, 1, 0, -49, 196, -294, 210, -77, 14, -1, 0, -64, 336, -672, 660, -352, 104, -16, 1, 0, -81, 540, -1386, 1782, -1287, 546, -135, 18, -1, 0, -100, 825, -2640, 4290, -4004, 2275, -800, 170, -20, 1

Offset: 1

Author

Eric W. Weisstein, Apr 06 2017

Keywords

• sign
• easy
• tabl

Comments

Extended to n = 1 and n = 2 using the closed form.

Closely related to A127677, which has opposite signs and rows begin with 2 of alternating signs instead of 0.

Examples

			First few polynomials are
-x
-4*x + x^2
-9*x + 6*x^2 - x^3
which give the triangle of coefficients:
0, -1;
0, -4, 1;
0, -9, 6, -1;
...
		

Links

• Eric Weisstein's World of Mathematics, Cycle Graph
• Eric Weisstein's World of Mathematics, Laplacian Polynomial

Crossrefs

Cf. A127677 (opposite signs and rows begin with +/2).

Programs

Mathematica

CoefficientList[LucasL[2 Range[10], Sqrt[-x]] - 2, x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[2 (ChebyshevT[Range[10], 1 - x/2] - 1), x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[FunctionExpand[2 (Cos[Range[10] ArcCos[1 - x/2]] - 1)], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[LinearRecurrence[{3 - x, -3 + x, 1}, {-x, (-4 + x) x, -(-3 + x)^2 x}, 10], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)