%I #50 Sep 04 2024 12:24:44
%S 1,-1,2,1,-8,8,-1,18,-48,32,1,-32,160,-256,128,-1,50,-400,1120,-1280,
%T 512,1,-72,840,-3584,6912,-6144,2048,-1,98,-1568,9408,-26880,39424,
%U -28672,8192,1,-128,2688,-21504,84480,-180224,212992,-131072,32768,-1,162,-4320,44352,-228096,658944,-1118208
%N Coefficient table for Chebyshev polynomials T(2*n,x) (increasing even powers x, without zeros).
%C Let C_n be the root lattice generated as a monoid by {+-2*e_i: 1 <= i <= n; +-e_i +- e_j: 1 <= i not equal to j <= n}. Let P(C_n) be the polytope formed by the convex hull of this generating set. Then the rows of (the signless version of) this array are the f-vectors of a unimodular triangulation of P(C_n) [Ardila et al.]. See A086645 for the corresponding array of h-vectors for these type C_n polytopes. See A063007 for the array of f-vectors for type A_n polytopes and A108556 for the array of f-vectors associated with type D_n polytopes. - _Peter Bala_, Oct 23 2008
%D Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990. p. 37, eq.(1.96) and p. 4. eq.(1.10).
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 795.
%H F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, <a href="http://arxiv.org/abs/0809.5123">Root polytopes and growth series of root lattices</a>, arXiv:0809.5123 [math.CO], 2008.
%H G. Boros, V. H. Moll, <a href="https://doi.org/10.10190/S0025-5718-01-01347-3">Landen transformations and the integration of rational functions</a>, Math. Comp. 71 (238) (2001) 649-668, absolute values in Lemma A.3.
%H C. Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a> (Annotated scans of selected pages) See p. 516.
%H Wolfdieter Lang, <a href="/A127674/a127674.pdf">Row polynomials.</a>
%H Richard J. Mathar, <a href="https://arxiv.org/abs/2408.15212">Chebyshev approximation of x^m (-log x)^l in the interval 0 <= x <= 1</a>, arXiv:2408.15212 [math.CA], 2024. See p. 1.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n,m) = 0 if n < m, a(0,0) = 1; otherwise a(n,m) = ((-1)^(n-m))*(2^(2*m-1))*binomial(n+m,2*m)*(2*n)/(n+m).
%F O.g.f.: (1 + z*(1 - 2*x))/((1 + z)^2 - 4*x*z) = 1 + (-1 + 2*x)*z + (1 - 8*x + 8*x^2)*z^2 + ... . [_Peter Bala_, Oct 23 2008] For the t-polynomials actually with x -> x^2. - _Wolfdieter Lang_, Aug 02 2014
%F Denoting the row polynomials by R(n,x) we have exp( Sum_{n >= 1} R(n,x)*z^n/n ) = 1/sqrt( (1 + z)^2 - 4*x*z ) = 1 + (-1 + 2*x)*z + (1 - 6*x + 6*x^2)*z^2 + ..., the o.g.f. for a signed version of A063007. - _Peter Bala_, Sep 02 2015
%F The n-th row polynomial equals T(n, 2*x - 1). - _Peter Bala_, Jul 09 2023
%e [1];
%e [-1,2];
%e [1,-8,8];
%e [-1,18,-48,32];
%e [1,-32,160,-256,128];
%e ...
%e See a link for the row polynomials.
%e The T-polynomial for row n=3, [-1,18,-48,32], is T(2*3,x) = -1*x^0 + 18*x^2 - 48*x^4 + 32*x^6.
%Y Cf. A075733 (different signs and offset). A084930 (coefficients of odd-indexed T-polynomials).
%Y Cf. A053120 (coefficients of T-polynomials, with interspersed zeros).
%Y Cf. A086645, A063007, A108556.
%K sign,easy,tabl
%O 0,3
%A _Wolfdieter Lang_ Mar 07 2007