A123583 Triangle read by rows: T(n, k) is the coefficient of x^k in the polynomial 1 - T_n(x)^2, where T_n(x) is the n-th Chebyshev polynomial of the first kind.

0, 1, 0, -1, 0, 0, 4, 0, -4, 1, 0, -9, 0, 24, 0, -16, 0, 0, 16, 0, -80, 0, 128, 0, -64, 1, 0, -25, 0, 200, 0, -560, 0, 640, 0, -256, 0, 0, 36, 0, -420, 0, 1792, 0, -3456, 0, 3072, 0, -1024, 1, 0, -49, 0, 784, 0, -4704, 0, 13440, 0, -19712, 0, 14336, 0, -4096

Offset

0,7

Comments

All row sum are zero. Row sums of absolute values are in A114619. - Klaus Brockhaus, May 29 2009

Links

G. C. Greubel, Rows n = 0..50, flattened

Gareth Jones and David Singerman, Belyi Functions, Hypermaps and Galois Groups, Bull. London Math. Soc., 28 (1996), 561-590.

Yuri Matiyasevich, Generalized Chebyshev polynomials.

G. B. Shabat and I. A. Voevodskii, Drawing curves over number fields, The Grothendieck Festschift, vol. 3, Birkhäuser, 1990, 199-227.

G. B. Shabat and A. Zvonkin, Plane trees and algebraic numbers, Contemporary Math., 1994, vol. 178, 233-275.

Formula

T(n, k) = coefficients of ( 1 - ChebyshevT(n, x)^2 ).

T(n, k) = coefficients of ( (1 - ChebyshevT(2*n, x))/2 ). - G. C. Greubel, Jul 02 2021

Example

First few rows of the triangle are:
  0;
  1, 0,  -1;
  0, 0,   4, 0,   -4;
  1, 0,  -9, 0,   24, 0,  -16;
  0, 0,  16, 0,  -80, 0,  128, 0,   -64;
  1, 0, -25, 0,  200, 0, -560, 0,   640, 0, -256;
  0, 0,  36, 0, -420, 0, 1792, 0, -3456, 0, 3072, 0, -1024;
First few polynomials (p(n, x) = 1 - T_{n}(x)^2) are:
  p(0, x) = 0,
  p(1, x) = 1 -    x^2,
  p(2, x) = 0    4*x^2 -   4*x^4,
  p(3, x) = 1 -  9*x^2 +  24*x^4 -   16*x^6,
  p(4, x) = 0   16*x^2 -  80*x^4 +  128*x^6 -   64*x^8,
  p(5, x) = 1 - 25*x^2 + 200*x^4 -  560*x^6 +  640*x^8 -  256*x^10,
  p(6, x) = 0   36*x^2 - 420*x^4 + 1792*x^6 - 3456*x^8 + 3072*x^10 - 1024*x^12.

Mathematica

(* First program *)
Table[CoefficientList[1 - ChebyshevT[n, x]^2, x], {n, 0, 10}]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k]= SeriesCoefficient[(1 -ChebyshevT[2*n, x])/2, {x, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, 2*n}]//Flatten (* G. C. Greubel, Jul 02 2021 *)

Prog

(Magma) [0] cat &cat[ Coefficients(1-ChebyshevT(n)^2): n in [1..8] ];
(PARI) v=[]; for(n=0, 8, v=concat(v, vector(2*n+1, j, polcoeff(1-poltchebi(n)^2, j-1)))); v
(Sage)
def T(n): return ( (1 - chebyshev_T(2*n, x))/2 ).full_simplify().coefficients(sparse=False)
flatten([T(n) for n in (0..12)]) # G. C. Greubel, Jul 02 2021

Crossrefs

Cf. A123588, A156647.

Sequence in context: A112919 A019201 A137660 * A236112 A226787 A140574

Adjacent sequences: A123580 A123581 A123582 * A123584 A123585 A123586

Keyword

tabf,sign

Author

Gary W. Adamson and Roger L. Bagula, Nov 12 2006

Extensions

Edited by N. J. A. Sloane, Mar 09 2008

Status

approved