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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)).
0
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).
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: A321514 A280306 A376511 * A106051 A066781 A112869
KEYWORD
tabf,uned,sign
AUTHOR
STATUS
approved