A114619 -id:A114619 - cp's OEIS frontend

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

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

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

			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.

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.

Cf. A123588, A156647.

[0] cat &cat[ Coefficients(1-ChebyshevT(n)^2): n in [1..8] ];

(* 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 *)

v=[]; for(n=0, 8, v=concat(v, vector(2*n+1, j, polcoeff(1-poltchebi(n)^2, j-1)))); v

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

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

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

A123588 Triangle read by rows: T(n, k) is the coefficient of x^k in the polynomial 1 - ChT(n, x^(1/2))^2, where ChT(n, x) is the n-th Chebyshev polynomial of the first kind, evaluated at x (0 <= k <= n).

Original entry on oeis.org

0, 1, -1, 0, 4, -4, 1, -9, 24, -16, 0, 16, -80, 128, -64, 1, -25, 200, -560, 640, -256, 0, 36, -420, 1792, -3456, 3072, -1024, 1, -49, 784, -4704, 13440, -19712, 14336, -4096, 0, 64, -1344, 10752, -42240, 90112, -106496, 65536, -16384, 1, -81, 2160, -22176, 114048, -329472, 559104, -552960, 294912

Offset: 0

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

			Polynomials:
0,
1 - x^2,
4 x^2 - 4 x^4,
1 - 9 x^2 + 24 x^4 - 16 x^6,
16 x^2 - 80 x^4 + 128 x^6 - 64 x^8,
1 - 25 x^2 + 200 x^4 - 560 x^6 + 640 x^8 - 256 x^10
Triangle starts:
  0;
  1,  -1;
  0,   4,  -4;
  1,  -9,  24,  -16;
  0,  16, -80,  128, -64;
  1, -25, 200, -560, 640, -256;

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

G. C. Greubel, Table of n, a(n) for the first 50 rows, 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 A. Zvonkin, Plane trees and algebraic numbers, Contemporary Math., 1994, vol. 178, 233-275.

Cf. A000004 (row sums vanish), A114619 (alternating row sums).

with(orthopoly): for n from 0 to 9 do seq(coeff(expand((1-T(n,sqrt(x))^2)),x,k), k=0..n) od; # yields sequence in triangular form

row[0] = {0}; row[n_] := CoefficientList[1 - ChebyshevT[n, x^(1/2)]^2, x]; Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Jan 29 2016 *)
T[n_,k_]:=If[k==0,Mod[n,2],(-1)^(n+k-1)*4^(k-1)*(2*Binomial[n+k,2*k]-Binomial[n+k-1,2*k-1])];Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Aug 11 2024 *)

T(n, 0) = (n mod 2); T(n,k) = (-1)^(n + k - 1)*4^(k - 1)*(2*binomial(n + k, 2*k) - binomial(n + k - 1, 2*k - 1)) for k > 0. - Detlef Meya, Aug 11 2024

Edited by N. J. A. Sloane, Dec 03 2006

cp's OEIS Frontend

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.

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