A140324 A new way to compute polynomial triangles from matrices of a Folium Implicit type: M={{0, -w[1], -w[2]}, {w[1], 0, -w[1]}, {w[2], w[1], 0}} that gives even only monomials as w[1]=x, others as one.

1, 0, 0, 1, 1, -2, -1, 2, 1, 1, -8, 22, -22, 1, 6, 1, 0, 0, 9, -54, 117, -102, 18, 12, 1, 1, -6, 3, 48, -101, -32, 291, -294, 70, 20, 1

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, May 26 2008

Keywords

• uned
• tabf
• sign

Comments

Matrix of the type

{{x,y,a},

{y,a,x},

{a,x,y}}

gives the folium of Descartes implicit polynomial:

x^3+y^3+a^3-3a*x*y

These types of polynomials gives various types of implicit curves in higher dimensions.

Unsigned version of this sequence algorithm gives A055137.

Some of these polynomials are similar to the Hodge number / diamond type Calabi-Yau implicit or Algebraic varieties. Here I have invented a way to make monomials from the higher polynomials. In the past I have used this matrix method to produce 3d Implicit surfaces.

Examples

			{1},
{},
{0, 0, 1},
{},
{1, -2, -1, 2, 1},
{},
{1, -8, 22, -22, 1, 6, 1},
{},
{0, 0, 9, -54, 117, -102, 18, 12, 1},
{},
{1, -6, 3, 48, -101, -32, 291, -294, 70, 20, 1}
		

Programs

Mathematica

Clear[M, a, d, x, w] M[d_] := Table[Sign[n - m]*w[Abs[n - m]], {n, 1, d}, {m, 1, d}]; a = Table[M[d], {d, 1, 10}]; Table[If[n == 1, w[n] = x, w[n] = 1], {n, 0, 10}]; Table[Det[a[[d]]], {d, 1, 10}]; a0 = Join[{{1}}, Table[CoefficientList[Det[a[[d]]], x], {d, 1, 10}]]; Flatten[a0] Table[Apply[Plus, CoefficientList[Det[a[[d]]], x]], {d, 1, 10}]

Formula

Compute matrices as: T(n,m)=Sign[n - m]*w[Abs[n - m]]; Change to monomial as:If[n==1,w[n]=x,w[n]=1]; Take determinant of matrices M(d); out_n,m=Coefficients(Det(M(d)))).