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
Examples
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)
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.
Programs
-
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 *)
Comments