OFFSET
1,6
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.
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)))).
EXAMPLE
{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}
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}]
CROSSREFS
KEYWORD
uned,tabf,sign
AUTHOR
Roger L. Bagula and Gary W. Adamson, May 26 2008
STATUS
approved