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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. 0

%I #11 Apr 28 2014 01:15:39

%S 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,

%T -6,3,48,-101,-32,291,-294,70,20,1

%N 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.

%C Matrix of the type

%C {{x,y,a},

%C {y,a,x},

%C {a,x,y}}

%C gives the folium of Descartes implicit polynomial:

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

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

%C Unsigned version of this sequence algorithm gives A055137.

%C 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.

%F 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)))).

%e {1},

%e {},

%e {0, 0, 1},

%e {},

%e {1, -2, -1, 2, 1},

%e {},

%e {1, -8, 22, -22, 1, 6, 1},

%e {},

%e {0, 0, 9, -54, 117, -102, 18, 12, 1},

%e {},

%e {1, -6, 3, 48, -101, -32, 291, -294, 70, 20, 1}

%t 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}]

%K uned,tabf,sign

%O 1,6

%A _Roger L. Bagula_ and _Gary W. Adamson_, May 26 2008

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Last modified February 22 06:21 EST 2024. Contains 370240 sequences. (Running on oeis4.)