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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.

%I A123583 #31 Sep 08 2022 08:45:28
%S A123583 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 A123583 -25,0,200,0,-560,0,640,0,-256,0,0,36,0,-420,0,1792,0,-3456,0,3072,0,
%U A123583 -1024,1,0,-49,0,784,0,-4704,0,13440,0,-19712,0,14336,0,-4096
%N 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.
%C A123583 All row sum are zero. Row sums of absolute values are in A114619. - _Klaus Brockhaus_, May 29 2009
%H A123583 G. C. Greubel, <a href="/A123583/b123583.txt">Rows n = 0..50, flattened</a>
%H A123583 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 A123583 Yuri Matiyasevich, <a href="http://logic.pdmi.ras.ru/~yumat/Journal/Chebyshev/chebysh.htm">Generalized Chebyshev polynomials</a>.
%H A123583 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 A123583 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 A123583 T(n, k) = coefficients of ( 1 - ChebyshevT(n, x)^2 ).
%F A123583 T(n, k) = coefficients of ( (1 - ChebyshevT(2*n, x))/2 ). - _G. C. Greubel_, Jul 02 2021
%e A123583 First few rows of the triangle are:
%e A123583   0;
%e A123583   1, 0,  -1;
%e A123583   0, 0,   4, 0,   -4;
%e A123583   1, 0,  -9, 0,   24, 0,  -16;
%e A123583   0, 0,  16, 0,  -80, 0,  128, 0,   -64;
%e A123583   1, 0, -25, 0,  200, 0, -560, 0,   640, 0, -256;
%e A123583   0, 0,  36, 0, -420, 0, 1792, 0, -3456, 0, 3072, 0, -1024;
%e A123583 First few polynomials (p(n, x) = 1 - T_{n}(x)^2) are:
%e A123583   p(0, x) = 0,
%e A123583   p(1, x) = 1 -    x^2,
%e A123583   p(2, x) = 0    4*x^2 -   4*x^4,
%e A123583   p(3, x) = 1 -  9*x^2 +  24*x^4 -   16*x^6,
%e A123583   p(4, x) = 0   16*x^2 -  80*x^4 +  128*x^6 -   64*x^8,
%e A123583   p(5, x) = 1 - 25*x^2 + 200*x^4 -  560*x^6 +  640*x^8 -  256*x^10,
%e A123583   p(6, x) = 0   36*x^2 - 420*x^4 + 1792*x^6 - 3456*x^8 + 3072*x^10 - 1024*x^12.
%t A123583 (* First program *)
%t A123583 Table[CoefficientList[1 - ChebyshevT[n, x]^2, x], {n, 0, 10}]//Flatten
%t A123583 (* Second program *)
%t A123583 T[n_, k_]:= T[n, k]= SeriesCoefficient[(1 -ChebyshevT[2*n,x])/2, {x,0,k}];
%t A123583 Table[T[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* _G. C. Greubel_, Jul 02 2021 *)
%o A123583 (Magma) [0] cat &cat[ Coefficients(1-ChebyshevT(n)^2): n in [1..8] ];
%o A123583 (PARI) v=[]; for(n=0, 8, v=concat(v, vector(2*n+1, j, polcoeff(1-poltchebi(n)^2, j-1)))); v
%o A123583 (Sage)
%o A123583 def T(n): return ( (1 - chebyshev_T(2*n, x))/2 ).full_simplify().coefficients(sparse=False)
%o A123583 flatten([T(n) for n in (0..12)]) # _G. C. Greubel_, Jul 02 2021
%Y A123583 Cf. A123588, A156647.
%K A123583 tabf,sign
%O A123583 0,7
%A A123583 _Gary W. Adamson_ and _Roger L. Bagula_, Nov 12 2006
%E A123583 Edited by _N. J. A. Sloane_, Mar 09 2008