A143576 A triangle of coefficients formed by adding Chebyshev T(x,n) to x*U(x,n): p(x,n)=ChebyshevT[n, x] + x*ChebyshevU[n, x].

1, 1, 1, 0, 1, 2, -1, -1, 2, 4, 0, -3, -4, 4, 8, 1, 1, -8, -12, 8, 16, 0, 5, 6, -20, -32, 16, 32, -1, -1, 18, 24, -48, -80, 32, 64, 0, -7, -8, 56, 80, -112, -192, 64, 128, 1, 1, -32, -40, 160, 240, -256, -448, 128, 256, 0, 9, 10, -120, -160, 432, 672, -576, -1024, 256, 512, -1, -1, 50, 60, -400, -560, 1120, 1792, -1280, -2304, 512, 1024

Offset

0,6

Comments

Row sums are: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

The idea of joining the cosine like Chebyshev to the sine like in this manner is to get rid of the most of the zeros.

A {1} was added to the top of the triangle to make it complete with row sums starting at 1.

Links

Table of n, a(n) for n=0..77.

Formula

p(x,n)=ChebyshevT[n, x] + x*ChebyshevU[n, x]; t(n,m)=Coefficients(p(x,n)).

Example

{1},
{1, 1},
{0, 1, 2},
{-1, -1, 2, 4},
{0, -3, -4, 4, 8},
{1, 1, -8, -12, 8, 16},
{0, 5, 6, -20, -32, 16, 32},
{-1, -1, 18, 24, -48, -80,32, 64},
{0, -7, -8, 56, 80, -112, -192, 64, 128},
{1, 1, -32, -40, 160, 240, -256, -448, 128, 256},
{0, 9, 10, -120, -160, 432, 672, -576, -1024, 256, 512},
{-1, -1, 50, 60, -400, -560, 1120, 1792, -1280, -2304, 512, 1024}

Mathematica

Clear[p, x, n, a]; p[x_, n_] = ChebyshevT[n, x] + x*ChebyshevU[n, x]; Table[FullSimplify[Expand[p[x, n]]], {n, 0, 10}]; Join[{{1}}, Table[CoefficientList[FullSimplify[Expand[p[x, n]]], x], {n, 0, 10}]]; Flatten[%]

Crossrefs

Cf. A053120.

Sequence in context: A030018 A010739 A166918 * A297159 A293438 A318622

Adjacent sequences: A143573 A143574 A143575 * A143577 A143578 A143579

Keyword

uned,sign,tabl,less

Author

Roger L. Bagula and Gary W. Adamson, Oct 25 2008

Status

approved