cp's OEIS Frontend

A028297 Coefficients of Chebyshev polynomials of the first kind: triangle of coefficients in expansion of cos(n*x) in descending powers of cos(x).

%I A028297 #111 Jun 27 2025 01:15:08
%S A028297 1,1,2,-1,4,-3,8,-8,1,16,-20,5,32,-48,18,-1,64,-112,56,-7,128,-256,
%T A028297 160,-32,1,256,-576,432,-120,9,512,-1280,1120,-400,50,-1,1024,-2816,
%U A028297 2816,-1232,220,-11,2048,-6144,6912,-3584,840,-72,1,4096,-13312,16640,-9984
%N A028297 Coefficients of Chebyshev polynomials of the first kind: triangle of coefficients in expansion of cos(n*x) in descending powers of cos(x).
%C A028297 Rows are of lengths 1, 1, 2, 2, 3, 3, ... (A008619).
%C A028297 This triangle is generated from A118800 by shifting down columns to allow for (1, 1, 2, 2, 3, 3, ...) terms in each row. - _Gary W. Adamson_, Dec 16 2007
%C A028297 Unsigned triangle = A034839 * A007318. - _Gary W. Adamson_, Nov 28 2008
%C A028297 Triangle, with zeros omitted, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, -1, 1, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Dec 16 2011
%C A028297 From _Wolfdieter Lang_, Aug 02 2014: (Start)
%C A028297 This irregular triangle is the row reversed version of the Chebyshev T-triangle A053120 given by A039991 with vanishing odd-indexed columns removed.
%C A028297 If zeros are appended in each row n >= 1, in order to obtain a regular triangle (see the _Philippe Deléham_ comment, g.f. and example) this becomes the Riordan triangle (1-x)/(1-2*x), -x^2/(1-2*x). See also the unsigned version A201701 of this regular triangle.
%C A028297 (End)
%C A028297 Apparently, unsigned diagonals of this array are rows of A200139. - _Tom Copeland_, Oct 11 2014
%C A028297 It appears that the coefficients are generated by the following: Let SM_k = Sum( d_(t_1, t_2)* eM_1^t_1 * eM_2^t_2) summed over all length 2 integer partitions of k, i.e., 1*t_1 + 2*t_2 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 2 data (i.e., SM_k = S_k/2 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(2,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2) form an irregular triangle, with one row for each k value starting with k=1. Thus this procedure and associated OEIS sequences A287768, A288199, A288207, A288211, A288245, A288188 are generalizations of Chebyshev polynomials of the first kind. - _Gregory Gerard Wojnar_, Jul 01 2017
%D A028297 I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed., Section 1.335, p. 35.
%D A028297 S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 106. [From _Rick L. Shepherd_, Jul 06 2010]
%H A028297 Alois P. Heinz, <a href="/A028297/b028297.txt">Rows n = 0..200, flattened</a>
%H A028297 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] p. 795.
%H A028297 Pantelis A. Damianou, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.121.02.120">A Beautiful Sine Formula</a>, Amer. Math. Monthly 121 (2014), no. 2, 120--135. MR3149030.
%H A028297 Daniel J. Greenhoe, <a href="https://www.researchgate.net/publication/337858762_Frames_and_Bases_Structure_and_Design_version_020">Frames and Bases: Structure and Design</a>, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, see page 172.
%H A028297 Daniel J. Greenhoe, <a href="https://www.researchgate.net/publication/337858659_A_Book_Concerning_Transforms_version_010">A Book Concerning Transforms</a>, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 94.
%H A028297 Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, <a href="https://doi.org/10.3934/era.2020057">Recursive sequences and Girard-Waring identities with applications in sequence transformation</a>, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
%H A028297 C. Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a> (Annotated scans of selected pages)
%H A028297 G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, <a href="https://arxiv.org/abs/1706.08381">Universal Peculiar Linear Mean Relationships in All Polynomials</a>, Table GW.n=2, p. 22, arXiv:1706.08381 [math.GM], 2017.
%F A028297 cos(n*x) = 2 * cos((n-1)*x) * cos(x) - cos((n-2)*x) (from CRC's Multiple-angle relations). - _Rick L. Shepherd_, Jul 06 2010
%F A028297 G.f.: (1-x) / (1-2x+y*x^2). - _Philippe Deléham_, Dec 16 2011
%F A028297 Sum_{k=0..n} T(n,k)*x^k = A011782(n), A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x = 0, 1, 2, 3, 4, 5, 6, respectively. - _Philippe Deléham_, Dec 16 2011
%F A028297 T(n,k) = [x^k] hypergeom([1/2 - n/2, -n/2], [1/2], 1 - x). - _Peter Luschny_, Feb 03 2021
%F A028297 T(n,k) = (-1)^k * 2^(n-1-2*k) * A034807(n,k). - _Hoang Xuan Thanh_, Jun 21 2025
%e A028297 Letting c = cos x, we have: cos 0x = 1, cos 1x = 1c; cos 2x = 2c^2-1; cos 3x = 4c^3-3c, cos 4x = 8c^4-8c^2+1, etc.
%e A028297 T4 = 8x^4 - 8x^2 + 1 = 8, -8, +1 = 2^(3) - (4)(2) + [2^(-1)](4)/2.
%e A028297 From _Wolfdieter Lang_, Aug 02 2014: (Start)
%e A028297 The irregular triangle T(n,k) begins:
%e A028297 n\k     1      2     3      4     5     6   7   8 ....
%e A028297 0:      1
%e A028297 1:      1
%e A028297 2:      2     -1
%e A028297 3:      4     -3
%e A028297 4:      8     -8     1
%e A028297 5:     16    -20     5
%e A028297 6:     32    -48    18     -1
%e A028297 7:     64   -112    56     -7
%e A028297 8:    128   -256   160    -32     1
%e A028297 9:    256   -576   432   -120     9
%e A028297 10:   512  -1280  1120   -400    50    -1
%e A028297 11:  1024  -2816  2816  -1232   220   -11
%e A028297 12:  2048  -6144  6912  -3584   840   -72   1
%e A028297 13:  4096 -13312 16640  -9984  2912  -364  13
%e A028297 14:  8192 -28672 39424 -26880  9408 -1568  98  -1
%e A028297 15: 16384 -61440 92160 -70400 28800 -6048 560 -15
%e A028297 ...
%e A028297 T(4,x) = 8*x^4 -8*x^2 + 1*x^0, T(5,x) = 16*x^5 - 20*x^3 + 5*x^1, with Chebyshev's T-polynomials (A053120). (End)
%e A028297 From _Philippe Deléham_, Dec 16 2011: (Start)
%e A028297 The triangle (1,1,0,0,0,0,...) DELTA (0,-1,1,0,0,0,0,...) includes zeros and begins:
%e A028297    1;
%e A028297    1,   0;
%e A028297    2,  -1,  0;
%e A028297    4,  -3,  0,  0;
%e A028297    8,  -8,  1,  0, 0;
%e A028297   16, -20,  5,  0, 0, 0;
%e A028297   32, -48, 18, -1, 0, 0, 0; (End)
%p A028297 b:= proc(n) b(n):= `if`(n<2, 1, expand(2*b(n-1)-x*b(n-2))) end:
%p A028297 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
%p A028297 seq(T(n), n=0..15);  # _Alois P. Heinz_, Sep 04 2019
%t A028297 t[n_] := (Cos[n x] // TrigExpand) /. Sin[x]^m_ /; EvenQ[m] -> (1 - Cos[x]^2)^(m/2) // Expand; Flatten[Table[ r = Reverse @ CoefficientList[t[n], Cos[x]]; If[OddQ[Length[r]], AppendTo[r,0]]; Partition[r,2][[All, 1]],{n, 0, 13}] ][[1 ;; 53]] (* _Jean-François Alcover_, May 06 2011 *)
%t A028297 Tpoly[n_] := HypergeometricPFQ[{(1 - n)/2, -n/2}, {1/2}, 1 - x];
%t A028297 Table[CoefficientList[Tpoly[n], x], {n, 0, 12}] // Flatten (* _Peter Luschny_, Feb 03 2021 *)
%Y A028297 Cf. A028298.
%Y A028297 Reflection of A008310, the main entry. With zeros: A039991.
%Y A028297 Cf. A053120 (row reversed table including zeros).
%Y A028297 Cf. A118800, A034839, A081277, A124182.
%Y A028297 Cf. A001333 (row sums 1), A001333 (alternating row sums). - _Wolfdieter Lang_, Aug 02 2014
%K A028297 tabf,easy,sign
%O A028297 0,3
%A A028297 _N. J. A. Sloane_
%E A028297 More terms from _David W. Wilson_
%E A028297 Row length sequence and link to Abramowitz-Stegun added by _Wolfdieter Lang_, Aug 02 2014