This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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). 0
 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 (list; graph; refs; listen; history; text; internal format)
 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 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: A227962 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 8 01:45 EST 2019. Contains 329850 sequences. (Running on oeis4.)