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 A028297 Coefficients of Chebyshev polynomials of the first kind: triangle of coefficients in expansion of cos(n*x) in descending powers of cos(x). 24

%I

%S 1,1,2,-1,4,-3,8,-8,1,16,-20,5,32,-48,18,-1,64,-112,56,-7,128,-256,

%T 160,-32,1,256,-576,432,-120,9,512,-1280,1120,-400,50,-1,1024,-2816,

%U 2816,-1232,220,-11,2048,-6144,6912,-3584,840,-72,1,4096,-13312,16640,-9984

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

%C Rows are of lengths 1, 1, 2, 2, 3, 3, ... (A008619).

%C 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 Unsigned triangle = A034839 * A007318. - _Gary W. Adamson_, Nov 28 2008

%C 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 From _Wolfdieter Lang_, Aug 02 2014: (Start)

%C This irregular triangle is the row reversed version of the Chebyshev T-triangle A053120 given by A039991 with vanishing odd indexed columns removed.

%C 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 (End)

%C Apparently, unsigned diagonals of this array are rows of A200139. - _Tom Copeland_, Oct 11 2014

%C 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 Tchebysheb polynomials of the first kind. - _Gregory Gerard Wojnar_, Jul 01 2017

%D I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed., Section 1.335, p. 35.

%D S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 106. [From _Rick L. Shepherd_, Jul 06 2010]

%H Alois P. Heinz, <a href="/A028297/b028297.txt">Rows n = 0..200, flattened</a>

%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] p. 795.

%H Pantelis A. Damianou, <a href="http://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 C. Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a> (Annotated scans of selected pages)

%H G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, <a href="http://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 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 G.f.: (1-x)/(1-2x+y*x^2). - _Philippe Deléham_, Dec 16 2011

%F 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

%e 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 1; 1; 2,-1; 4,-3; 8,-8,1; 16,-20,5; 32,-48,18,-1; ...

%e T4 = 8x^4 - 8x^2 + 1 = 8, -8, +1 = 2^(3) - (4)(2) + [2^(-1)](4)/2.

%e From _Philippe Deléham_, Dec 16 2011: (Start)

%e Triangle (1,1,0,0,0,0,...) DELTA (0,-1,1,0,0,0,0,...) begins:

%e 1;

%e 1, 0;

%e 2, -1, 0;

%e 4, -3, 0, 0;

%e 8, -8, 1, 0, 0;

%e 16, -20, 5, 0, 0, 0;

%e 32, -48, 18, -1, 0, 0, 0; (End)

%e From _Wolfdieter Lang_, Aug 02 2014: (Start)

%e The irregular triangle t(n,k) begins:

%e n\k 1 2 3 4 5 6 7 8 ....

%e 0: 1

%e 1: 1

%e 2: 2 -1

%e 3: 4 -3

%e 4: 8 -8 1

%e 5: 16 -20 5

%e 6: 32 -48 18 -1

%e 7: 64 -112 56 -7

%e 8: 128 -256 160 -32 1

%e 9: 256 -576 432 -120 9

%e 10: 512 -1280 1120 -400 50 -1

%e 11: 1024 -2816 2816 -1232 220 -11

%e 12: 2048 -6144 6912 -3584 840 -72 1

%e 13: 4096 -13312 16640 -9984 2912 -364 13

%e 14: 8192 -28672 39424 -26880 9408 -1568 98 -1

%e 15: 16384 -61440 92160 -70400 28800 -6048 560 -15

%e ...

%e 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)

%t 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 *)

%Y Cf. A028298.

%Y Reflection of A008310, the main entry. With zeros: A039991.

%Y Cf. A053120 (row reversed table including zeros).

%Y Cf. A118800, A034839, A081277, A124182.

%Y Cf. A001333 (row sums 1), A001333 (alternating row sums). - _Wolfdieter Lang_, Aug 02 2014

%K tabf,easy,sign

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _David W. Wilson_

%E Row length sequence and link to Abramowitz-Stegun added by _Wolfdieter Lang_, Aug 02 2014

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Last modified November 14 21:49 EST 2018. Contains 317217 sequences. (Running on oeis4.)