A140723 A triangular sequence of coefficients of a truncated quotient (remainder dropped) of the ChebyshevT polynomials T(x,n) by the Cyclotomic polynomials c(x,n): p(x,n)=Quotient(T(x.n)/c(x,n)).

1, 1, -2, 2, -4, 4, -16, 0, 8, -16, 16, -14, -80, -48, 32, 32, -64, 64, 32, 0, -256, 0, 128, -256, -576, 0, 256, -912, 608, 1120, -1280, -1280, 512, 512

Offset

1,3

Comments

Row sums are:

{1, 1, 0, 0, -8, 0, -78, 0, -96, -576, -720};

Interesting effect here is that for primes:

p(x,n)=2^(Prime[n]-1)*(x-1).

Links

Table of n, a(n) for n=1..34.

Formula

ChebyshevT polynomials T(x,n): Cyclotomic polynomials c(x,n): p(x,n)=Quotient(T(x.n)/c(x,n)) a(n,m)=Coefficients(p(x,n)).

Example

{1},
{1},
{-2, 2},
{-4, 4},
{-16, 0, 8},
{-16, 16},
{-14, -80, -48, 32, 32},
{-64, 64},
{32, 0, -256, 0,128},
{-256, -576, 0, 256},
{-912, 608, 1120, -1280, -1280, 512, 512}

Mathematica

Clear[p, x, n, a] p[x_, n_] = PolynomialQuotient[ChebyshevT[n, x], Cyclotomic[n, x], x]; Table[p[x, n], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a]

Crossrefs

Sequence in context: A327486 A321514 A280306 * A106051 A066781 A112869

Adjacent sequences: A140720 A140721 A140722 * A140724 A140725 A140726

Keyword

tabf,uned,sign

Author

Roger L. Bagula and Gary W. Adamson, Jul 12 2008

Status

approved