A008312 Triangle of coefficients of Chebyshev polynomials U_n(x).

1, 2, -1, 4, -4, 8, 1, -12, 16, 6, -32, 32, -1, 24, -80, 64, -8, 80, -192, 128, 1, -40, 240, -448, 256, 10, -160, 672, -1024, 512, -1, 60, -560, 1792, -2304, 1024, -12, 280, -1792, 4608, -5120, 2048

Offset

0,2

Comments

Version with zeros in A053117. - Philippe Deléham, Nov 27 2013

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.

Links

T. D. Noe, Rows n = 0..100 of triangle, flattened

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

D. Foata and G.-N. Han, Nombres de Fibonacci et polynomes orthogonaux.

Valentin Ovsienko, Towards quantized complex numbers: q-deformed Gaussian integers and the Picard group, arXiv:2103.10800 [math.QA], 2021.

M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From N. J. A. Sloane, Feb 13 2013

Index entries for sequences related to Chebyshev polynomials.

Example

From Philippe Deléham, Nov 27 2013: (Start)
Triangle begins:
    1;
    2;
   -1,    4;
   -4,    8;
    1,  -12,    16;
    6,  -32,    32;
   -1,   24,   -80,    64;
   -8,   80,  -192,   128;
    1,  -40,   240,  -448,   256;
   10, -160,   672, -1024,   512;
   -1,   60,  -560,  1792, -2304, 1024;
  -12,  280, -1792,  4608, -5120, 2048;
  ...
With zeros, triangle begins:
   1;
   0,   2;
  -1,   0,   4;
   0,  -4,   0,    8;
   1,   0, -12,    0,   16;
   0,   6,   0,  -32,    0,    32;
  -1,   0,  24,    0,  -80,     0,   64;
   0,  -8,   0,   80,    0,  -192,    0,   128;
   1,   0, -40,    0,  240,     0, -448,     0,   256;
   0,  10,   0, -160,    0,   672,    0, -1024,     0,   512;
  -1,   0,  60,    0, -560,     0, 1792,     0, -2304,     0, 1024;
   0, -12,   0,  280,    0, -1792,    0,  4608,     0, -5120,    0, 2048;
  ...
(End)

Mathematica

a[n_, k_] := Coefficient[ ChebyshevU[n, x], x, k]; row[n_] := Table[a[n, k], {k, Mod[n, 2], n, 2}]; Table[row[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Oct 03 2012 *)

Crossrefs

Reflected version with zeros: A053118. Cf. A008310, A053112, A053117.

Sequence in context: A077965 A077967 A296188 * A345442 A060723 A300622

Adjacent sequences: A008309 A008310 A008311 * A008313 A008314 A008315

Keyword

sign,tabf,easy,nice

Author

N. J. A. Sloane

Status

approved