A137307 A triangular sequence of coefficients of even plus odd Chebyshev polynomials, A053120: q(x,n)=T(x,2*n-1]+T(x,2*n).

1, 1, -1, 1, 2, 1, -3, -8, 4, 8, -1, 5, 18, -20, -48, 16, 32, 1, -7, -32, 56, 160, -112, -256, 64, 128, -1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512, 1, -11, -72, 220, 840, -1232, -3584, 2816, 6912, -2816, -6144, 1024, 2048, -1, 13, 98, -364, -1568, 2912, 9408, -9984, -26880, 16640, 39424, -13312, -28672

Offset

1,5

Comments

The row sums are all 2 and double integrations are all orthogonal except for the zero to one level.

This arose from an idea of Chladni Chebyshev's:

q(Exp[i*t],n)=T(Cos[2*Pi*t),2*n-1)+T(Sin(2*Pi*t),2*n)

which are strange looping spirals.

Links

Table of n, a(n) for n=1..63.

Formula

q(x,n)=T(x,2*n-1]+T(x,2*n).

Example

{1, 1},
{-1, 1, 2},
{1, -3, -8, 4, 8},
{-1, 5,18, -20, -48, 16, 32},
{1, -7, -32, 56, 160, -112, -256, 64, 128},
{-1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512},
{1, -11, -72, 220, 840, -1232, -3584, 2816, 6912, -2816, -6144, 1024, 2048},
{-1, 13, 98, -364, -1568, 2912, 9408, -9984, -26880, 16640, 39424, -13312, -28672, 4096, 8192},
{1, -15, -128, 560, 2688, -6048, 21504, 28800, 84480, -70400, -180224, 92160, 212992, -61440, -131072, 16384, 32768},
{-1, 17, 162, -816, -4320, 11424, 44352, -71808, -228096, 239360, 658944, -452608, -1118208, 487424, 1105920, -278528, -589824, 65536, 131072},
{1, -19, -200, 1140, 6600, -20064, -84480, 160512, 549120, -695552, -2050048, 1770496, 4659200, -2723840, -6553600, 2490368, 5570560, -1245184, -2621440, 262144, 524288}

Mathematica

Q[x_, n_] := ChebyshevT[2*n - 1, x] + ChebyshevT[2*n, x]; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a0 = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a0]

Crossrefs

Cf. A053120.

Sequence in context: A248354 A260142 A194505 * A256420 A205391 A352858

Adjacent sequences: A137304 A137305 A137306 * A137308 A137309 A137310

Keyword

uned,tabl,sign

Author

Roger L. Bagula, Apr 20 2008

Status

approved