%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^33a*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 CalabiYau 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
