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Search: id:A140324
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| A140324 |
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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. |
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+0
1
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| 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
(list; graph; listen)
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OFFSET
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1,6
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COMMENT
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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.
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FORMULA
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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)))).
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EXAMPLE
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{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}
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MATHEMATICA
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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}]
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CROSSREFS
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Adjacent sequences: A140321 A140322 A140323 this_sequence A140325 A140326 A140327
Sequence in context: A003417 A079900 A117354 this_sequence A010250 A060024 A143668
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KEYWORD
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uned,tabf,sign
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AUTHOR
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Roger Bagula and Gary W. Adamsom (rlbagulatftn(AT)yahoo.com), May 26 2008
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