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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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%I #31 Sep 08 2022 08:45:28

%S 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,

%T -25,0,200,0,-560,0,640,0,-256,0,0,36,0,-420,0,1792,0,-3456,0,3072,0,

%U -1024,1,0,-49,0,784,0,-4704,0,13440,0,-19712,0,14336,0,-4096

%N 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.

%C All row sum are zero. Row sums of absolute values are in A114619. - _Klaus Brockhaus_, May 29 2009

%H G. C. Greubel, <a href="/A123583/b123583.txt">Rows n = 0..50, flattened</a>

%H Gareth Jones and David Singerman, <a href="https://doi.org/10.1112/blms/28.6.561">Belyi Functions, Hypermaps and Galois Groups</a>, Bull. London Math. Soc., 28 (1996), 561-590.

%H Yuri Matiyasevich, <a href="http://logic.pdmi.ras.ru/~yumat/Journal/Chebyshev/chebysh.htm">Generalized Chebyshev polynomials</a>.

%H G. B. Shabat and I. A. Voevodskii, <a href="https://www.math.ias.edu/Voevodsky/files/files-annotated/Dropbox/Published%20versions/drawing_curves_published.pdf">Drawing curves over number fields</a>, The Grothendieck Festschift, vol. 3, Birkhäuser, 1990, 199-227.

%H G. B. Shabat and A. Zvonkin, <a href="https://www.labri.fr/perso/zvonkin/Research/shabzvon.pdf">Plane trees and algebraic numbers</a>, Contemporary Math., 1994, vol. 178, 233-275.

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

%F T(n, k) = coefficients of ( (1 - ChebyshevT(2*n, x))/2 ). - _G. C. Greubel_, Jul 02 2021

%e First few rows of the triangle are:

%e 0;

%e 1, 0, -1;

%e 0, 0, 4, 0, -4;

%e 1, 0, -9, 0, 24, 0, -16;

%e 0, 0, 16, 0, -80, 0, 128, 0, -64;

%e 1, 0, -25, 0, 200, 0, -560, 0, 640, 0, -256;

%e 0, 0, 36, 0, -420, 0, 1792, 0, -3456, 0, 3072, 0, -1024;

%e First few polynomials (p(n, x) = 1 - T_{n}(x)^2) are:

%e p(0, x) = 0,

%e p(1, x) = 1 - x^2,

%e p(2, x) = 0 4*x^2 - 4*x^4,

%e p(3, x) = 1 - 9*x^2 + 24*x^4 - 16*x^6,

%e p(4, x) = 0 16*x^2 - 80*x^4 + 128*x^6 - 64*x^8,

%e p(5, x) = 1 - 25*x^2 + 200*x^4 - 560*x^6 + 640*x^8 - 256*x^10,

%e p(6, x) = 0 36*x^2 - 420*x^4 + 1792*x^6 - 3456*x^8 + 3072*x^10 - 1024*x^12.

%t (* First program *)

%t Table[CoefficientList[1 - ChebyshevT[n, x]^2, x], {n, 0, 10}]//Flatten

%t (* Second program *)

%t T[n_, k_]:= T[n, k]= SeriesCoefficient[(1 -ChebyshevT[2*n,x])/2, {x,0,k}];

%t Table[T[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* _G. C. Greubel_, Jul 02 2021 *)

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

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

%o (Sage)

%o def T(n): return ( (1 - chebyshev_T(2*n, x))/2 ).full_simplify().coefficients(sparse=False)

%o flatten([T(n) for n in (0..12)]) # _G. C. Greubel_, Jul 02 2021

%Y Cf. A123588, A156647.

%K tabf,sign

%O 0,7

%A _Gary W. Adamson_ and _Roger L. Bagula_, Nov 12 2006

%E Edited by _N. J. A. Sloane_, Mar 09 2008