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%I #9 Dec 01 2016 21:30:33
%S 3,1,2,-1,0,4,1,-6,0,8,3,0,-16,0,16,1,10,0,-40,0,32,-1,0,36,0,-96,0,
%T 64,1,-14,0,112,0,-224,0,128,3,0,-64,0,320,0,-512,0,256,1,18,0,-240,0,
%U 864,0,-1152,0,512,-1,0,100,0,-800,0,2240,0,-2560,0,1024
%N A triangle of coefficients of a Chebyshev T(x,n) polynomials to make pair binomials by in {x,y,z} and x only polynomial reduced: f(x,y,n)=Sum[CoefficientList[ChebyshevT[n, x], x][[i + 1]]*x^i*y^(n - i), {i,0, Length[CoefficientList[ChebyshevT[n, x], x]] - 1}]; p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(z,x,n).
%C Row sums are three:
%C The Algebraic varieties are projections of the Chebyshev orthogonal polynomials on interesting 3 dimensional implicit surfaces:
%C V(x,y,z,n)=p(x,y,z,n)-1:
%F f(x,y,n)=Sum[CoefficientList[ChebyshevT[n, x], x][[i + 1]]*x^i*y^(n - i), {i,0, Length[CoefficientList[ChebyshevT[n, x], x]] - 1}]; p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(z,x,n); Out_n,m=Coefficients(p(x,1,1,n).
%e {3},
%e {1, 2},
%e {-1, 0, 4},
%e {1, -6, 0, 8},
%e {3, 0, -16, 0, 16},
%e {1, 10, 0, -40, 0, 32},
%e {-1, 0, 36, 0, -96, 0, 64},
%e {1, -14, 0, 112, 0, -224, 0,128},
%e {3, 0, -64, 0, 320, 0, -512, 0, 256},
%e {1, 18, 0, -240, 0, 864, 0, -1152, 0, 512},
%e {-1, 0, 100, 0, -800, 0, 2240, 0, -2560, 0, 1024}
%e Polynomials:
%e 3,
%e 2 x + y,
%e 4 x^2 + y^2 - 2 z^2,
%e 8 x^3 - 3 x y^2 + 4 y^3 - 3 x z^2 - 3 y z^2,
%e 16 x^4 - 8 x^2 y^2 + 9 y^4 - 8 x^2 z^2 - 8 y^2 z^2 + 2 z^4
%t Clear[f, x, n] f[x_, y_, n_] := Sum[CoefficientList[ChebyshevT[n, x], x][[i + 1]]*x^i*y^(n - i), {i, 0, Length[CoefficientList[ChebyshevT[n, x], x]] - 1}]; Table[ExpandAll[f[x, y, n] + f[y, z, n] + f[x, z, n]], {n, 0, 10}]; a = Table[CoefficientList[ExpandAll[f[x, y, n] + f[y, z, n] + f[x, z, n]] /. y -> 1 /. z -> 1, x], {n, 0, 10}]; Flatten[a]
%t T[ n_, k_] := Coefficient[ 2 ChebyshevT[ n, x] + 1, x, k]; (* _Michael Somos_, Dec 01 2016 *)
%Y Cf. A053120.
%K uned,sign
%O 1,1
%A _Roger L. Bagula_ and _Gary W. Adamson_, Jun 11 2008
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