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A053120 Triangle of coefficients of Chebyshev's T(n,x) polynomials (powers of x in increasing order).

%I A053120 #127 Jan 08 2025 12:18:30
%S A053120 1,0,1,-1,0,2,0,-3,0,4,1,0,-8,0,8,0,5,0,-20,0,16,-1,0,18,0,-48,0,32,0,
%T A053120 -7,0,56,0,-112,0,64,1,0,-32,0,160,0,-256,0,128,0,9,0,-120,0,432,0,
%U A053120 -576,0,256,-1,0,50,0,-400,0,1120,0,-1280,0,512,0,-11,0,220,0,-1232,0,2816,0,-2816,0,1024
%N A053120 Triangle of coefficients of Chebyshev's T(n,x) polynomials (powers of x in increasing order).
%C A053120 Row sums (signed triangle): A000012 (powers of 1). Row sums (unsigned triangle): A001333(n).
%C A053120 From _Wolfdieter Lang_, Oct 21 2013: (Start)
%C A053120 The row polynomials T(n,x) equal (S(n,2*x) - S(n-2,2*x))/2, n >= 0, with the row polynomials S from A049310, with S(-1,x) = 0, and S(-2,x) = -1.
%C A053120 The zeros of T(n,x) are x(n,k) = cos((2*k+1)*Pi/(2*n)), k = 0, 1, ..., n-1, n >= 1. (End)
%C A053120 From _Wolfdieter Lang_, Jan 03 2020 and _Paul Weisenhorn_: (Start)
%C A053120 The (sub)diagonal sequences {D_{2*k}(m)}_{m >= 0}, for k >= 0, have o.g.f. GD_{2*k}(x) = (-1)^k*(1-x)/(1-2*x)^(k+1), for k >= 0, and  GD_{2*k+1}(x) = 0, for k >= 0. This follows from their o.g.f. GGD(z, x) := Sum_{k>=0} GD_k(x)*z^n which is obtained from the o.g.f. of the T-triangle GT(z, x) = (1-x*z)/(1 - 2*x + z^2) (see the formula section) by GGD(z, x) = GT(z, x/z).
%C A053120 The explicit form is then D_{2*k}(m) = (-1)^k, for m = 0, and
%C A053120   (-1)^k*(2*k+m)*2^(m-1)*risefac(k+1, m-1)/m!, for m >= 1, with the rising factorial risefac(x, n). (End)
%D A053120 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964. Tenth printing, Wiley, 2002 (also electronically available), p. 795.
%D A053120 F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 pp. 77, 105.
%D A053120 Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
%D A053120 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 22, page 196.
%D A053120 TableCurve 2D, Automated curve fitting and equation discovery, Version 5.01 for Windows, User's Manual, Chebyshev Series Polynomials and Rationals, pages 12-21 - 12-24, SYSTAT Software, Inc., Richmond, WA, 2002.
%H A053120 T. D. Noe, <a href="/A053120/b053120.txt">Rows 0 to 100 of triangle, flattened</a>
%H A053120 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 [scanned copy], p. 795.
%H A053120 Paul Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry5/barry96s.html">Meixner-Type Results for Riordan Arrays and Associated Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.9.4, section 5.
%H A053120 Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">Addendum to Elliptic Lie Triad</a>.
%H A053120 P. Damianou, <a href="http://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv preprint arXiv:1110.6620 [math.RT], 2014. - From _Tom Copeland_, Oct 11 2014
%H A053120 Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%H A053120 Wolfdieter Lang, <a href="/A053120/a053120.pdf">Rows n = 0..20</a>.
%H A053120 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 16.
%H A053120 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.
%H A053120 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A053120 T(n, m) = A039991(n, n-m).
%F A053120 G.f. for row polynomials T(n,x) (signed triangle): (1-x*z)/(1-2*x*z+z^2). If unsigned: (1-x*z)/(1-2*x*z-z^2).
%F A053120 T(n, m) := 0 if n < m or n+m odd; T(n, m) = (-1)^(n/2) if m=0 (n even); otherwise T(n, m) = ((-1)^((n+m)/2 + m))*(2^(m-1))*n*binomial((n+m)/2-1, m-1)/m.
%F A053120 Recursion for n >= 2: T(n, m) = T*a(n-1, m-1) - T(n-2, m), T(n, m)=0 if n < m, T(n, -1) := 0, T(0, 0) = T(1, 1) = 1.
%F A053120 G.f. for m-th column (signed triangle): 1/(1+x^2) if m=0, otherwise (2^(m-1))*(x^m)*(1-x^2)/(1+x^2)^(m+1).
%F A053120 From _G. C. Greubel_, Aug 10 2022: (Start)
%F A053120 Sum_{k=0..floor(n/2)} T(n-k, k) = A000007(n).
%F A053120 T(2*n, n) = i^n * A036909(n/2) * (1+(-1)^n)/2 + [n=0]/3. (End)
%F A053120 T(n, k) = [x^k] T(n, x) for n >= 1, where T(n, x) = Sum_{k=1..n}(-1)^(n - k)*(n/ (2*k))*binomial(k, n - k)*(2*x)^(2*k - n). - _Peter Luschny_, Sep 20 2022
%e A053120 The triangle a(n,m) begins:
%e A053120 n\m  0  1   2    3     4    5     6     7      8    9   10...
%e A053120 0:   1
%e A053120 1:   0  1
%e A053120 2:  -1  0   2
%e A053120 3:   0 -3   0    4
%e A053120 4:   1  0  -8    0     8
%e A053120 5:   0  5   0  -20     0   16
%e A053120 6:  -1  0  18    0   -48    0    32
%e A053120 7:   0 -7   0   56     0 -112     0    64
%e A053120 8:   1  0 -32    0   160    0  -256     0    128
%e A053120 9:   0  9   0 -120     0  432     0  -576      0  256
%e A053120 10: -1  0  50    0  -400    0  1120     0  -1280    0  512
%e A053120 ... Reformatted and extended - _Wolfdieter Lang_, Oct 21 2013
%e A053120 E.g., the fourth row (n=3) corresponds to the polynomial T(3,x) = -3*x + 4*x^3.
%p A053120 with(orthopoly) ;
%p A053120 A053120 := proc(n,k)
%p A053120     T(n,x) ;
%p A053120     coeftayl(%,x=0,k) ;
%p A053120 end proc: # _R. J. Mathar_, Jun 30 2013
%p A053120 T := (n, x) -> `if`(n = 0, 1, add((-1)^(n - k) * (n/(2*k))*binomial(k, n - k) *(2*x)^(2*k - n), k = 1 ..n)):
%p A053120 seq(seq(coeff(T(n, x), x, k), k = 0..n), n = 0..11); # _Peter Luschny_, Sep 20 2022
%t A053120 t[n_, k_] := Coefficient[ ChebyshevT[n, x], x, k]; Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* _Jean-François Alcover_, Jan 16 2012 *)
%o A053120 (Magma) &cat[ Coefficients(ChebyshevT(n)): n in [0..11] ]; // _Klaus Brockhaus_, Mar 08 2008
%o A053120 (PARI) for(n=0,5,P=polchebyshev(n);for(k=0,n,print1(polcoeff(P,k)", "))) \\ _Charles R Greathouse IV_, Jan 16 2012
%o A053120 (Julia)
%o A053120 using Nemo
%o A053120 function A053120Row(n)
%o A053120     R, x = PolynomialRing(ZZ, "x")
%o A053120     p = chebyshev_t(n, x)
%o A053120     [coeff(p, j) for j in 0:n] end
%o A053120 for n in 0:6 A053120Row(n) |> println end # _Peter Luschny_, Mar 13 2018
%o A053120 (SageMath)
%o A053120 def f(n,k): # f = A039991
%o A053120     if (n<2 and k==0): return 1
%o A053120     elif (k<0 or k>n): return 0
%o A053120     else: return 2*f(n-1, k) - f(n-2, k-2)
%o A053120 def A053120(n,k): return f(n, n-k)
%o A053120 flatten([[A053120(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Aug 10 2022
%Y A053120 Cf. A000007, A000012, A001333, A036909, A039991.
%Y A053120 The first nonzero (sub)diagonal sequences are A011782, -A001792, A001793(n+1), -A001794, A006974, -A006975, A006976, -A209404.
%K A053120 sign,tabl,nice,easy
%O A053120 0,6
%A A053120 _Wolfdieter Lang_