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, 4096, 8192

Offset: 1

Author

Roger L. Bagula, Apr 20 2008

Keywords

• uned
• tabf
• sign

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.

Examples

			Triangle begins:
  {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},
  ...
		

Crossrefs

Cf. A053120.

Programs

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]

Formula

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