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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.

%I A140324 #11 Apr 28 2014 01:15:39
%S A140324 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 A140324 -6,3,48,-101,-32,291,-294,70,20,1
%N 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.
%C A140324 Matrix of the type
%C A140324 {{x,y,a},
%C A140324 {y,a,x},
%C A140324 {a,x,y}}
%C A140324 gives the folium of Descartes implicit polynomial:
%C A140324 x^3+y^3+a^3-3a*x*y
%C A140324 These types of polynomials gives various types of implicit curves in higher dimensions.
%C A140324 Unsigned version of this sequence algorithm gives A055137.
%C A140324 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 A140324 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 A140324 {1},
%e A140324 {},
%e A140324 {0, 0, 1},
%e A140324 {},
%e A140324 {1, -2, -1, 2, 1},
%e A140324 {},
%e A140324 {1, -8, 22, -22, 1, 6, 1},
%e A140324 {},
%e A140324 {0, 0, 9, -54, 117, -102, 18, 12, 1},
%e A140324 {},
%e A140324 {1, -6, 3, 48, -101, -32, 291, -294, 70, 20, 1}
%t A140324 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 A140324 uned,tabf,sign
%O A140324 1,6
%A A140324 _Roger L. Bagula_ and _Gary W. Adamson_, May 26 2008