A139569 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).

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, 64, 1, -14, 0, 112, 0, -224, 0, 128, 3, 0, -64, 0, 320, 0, -512, 0, 256, 1, 18, 0, -240, 0, 864, 0, -1152, 0, 512, -1, 0, 100, 0, -800, 0, 2240, 0, -2560, 0, 1024

Offset

1,1

Comments

Row sums are three:

The Algebraic varieties are projections of the Chebyshev orthogonal polynomials on interesting 3 dimensional implicit surfaces:

V(x,y,z,n)=p(x,y,z,n)-1:

Links

Table of n, a(n) for n=1..66.

Formula

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).

Example

{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, 64},
{1, -14, 0, 112, 0, -224, 0,128},
{3, 0, -64, 0, 320, 0, -512, 0, 256},
{1, 18, 0, -240, 0, 864, 0, -1152, 0, 512},
{-1, 0, 100, 0, -800, 0, 2240, 0, -2560, 0, 1024}
Polynomials:
3,
2 x + y,
4 x^2 + y^2 - 2 z^2,
8 x^3 - 3 x y^2 + 4 y^3 - 3 x z^2 - 3 y z^2,
16 x^4 - 8 x^2 y^2 + 9 y^4 - 8 x^2 z^2 - 8 y^2 z^2 + 2 z^4

Mathematica

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[ n_, k_] := Coefficient[ 2 ChebyshevT[ n, x] + 1, x, k]; (* Michael Somos, Dec 01 2016 *)

Crossrefs

Cf. A053120.

Sequence in context: A331105 A255615 A056931 * A201590 A235358 A086249

Adjacent sequences: A139566 A139567 A139568 * A139570 A139571 A139572

Keyword

uned,sign

Author

Roger L. Bagula and Gary W. Adamson, Jun 11 2008

Status

approved