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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
 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; refs; listen; history; text; internal format)
 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. LINKS Table of n, a(n) for n=1..36. 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 Sequence in context: A079900 A188317 A117354 * A323284 A010250 A268041 Adjacent sequences: A140321 A140322 A140323 * A140325 A140326 A140327 KEYWORD uned,tabf,sign AUTHOR Roger L. Bagula and Gary W. Adamson, May 26 2008 STATUS approved

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Last modified December 7 03:04 EST 2023. Contains 367622 sequences. (Running on oeis4.)