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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). 16
1, 1, 2, -1, 4, -3, 8, -8, 1, 16, -20, 5, 32, -48, 18, -1, 64, -112, 56, -7, 128, -256, 160, -32, 1, 256, -576, 432, -120, 9, 512, -1280, 1120, -400, 50, -1, 1024, -2816, 2816, -1232, 220, -11, 2048, -6144, 6912, -3584, 840, -72, 1, 4096, -13312, 16640, -9984 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Rows are of length 1, 1, 2, 2, 3, 3, ...

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

Unsigned = A034839 * A007318. - Gary W. Adamson, Nov 28 2008

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

REFERENCES

Damianou, Pantelis A., A Beautiful Sine Formula. Amer. Math. Monthly 121 (2014), no. 2, 120--135. MR3149030

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

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 106. [From Rick L. Shepherd, Jul 06 2010]

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

FORMULA

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

G.f.: (1-x)/(1-2x+y*x^2). - Philippe Deléham, Dec 16 2011

Sum_{k, 0<=k<=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, resspectively. - Philippe Deléham, Dec 16 2011

EXAMPLE

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.

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

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

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

1

1, 0

2, -1, 0

4, -3, 0, 0

8, -8, 1, 0, 0

16, -20, 5, 0, 0, 0

32, -48, 18, -1, 0, 0, 0 - Philippe Deléham, Dec 16 2011

MATHEMATICA

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 *)

CROSSREFS

Cf. A028298.

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

Cf. A053120 (table including zeros).

Cf. A118800.

A034839. - Gary W. Adamson, Nov 28 2008

Cf. A081277, A124182.

Sequence in context: A100818 A005291 A106624 * A207537 A114438 A238757

Adjacent sequences:  A028294 A028295 A028296 * A028298 A028299 A028300

KEYWORD

tabf,easy,sign

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

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Last modified April 25 03:36 EDT 2014. Contains 240994 sequences.