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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 lengths 1, 1, 2, 2, 3, 3, ... (A008619).

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 triangle = 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

From Wolfdieter Lang, Aug 02 2014: (Start)

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

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.

(End)

REFERENCES

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

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy] p. 795.

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

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, respectively. - 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

From Wolfdieter Lang, Aug 02 2014: (Start)

The irregular triangle t(n,k) begins:

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

0:      1

1:      1

2:      2     -1

3:      4     -3

4:      8     -8     1

5:     16    -20     5

6:     32    -48    18     -1

7:     64   -112    56     -7

8:    128   -256   160    -32     1

9:    256   -576   432   -120     9

10:   512  -1280  1120   -400    50    -1

11:  1024  -2816  2816  -1232   220   -11

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

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

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

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

...

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 (A53120) (End)

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 (row reversed table including zeros).

Cf. A118800, A034839, A081277, A124182.

Cf. A001333 (row sums 1), A001333 (alternating row sums). - Wolfdieter Lang, Aug 02 2014

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

Row length sequence and link to Abramowitz-Stegun added. Wolfdieter Lang, Aug 02 2014

STATUS

approved

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Last modified September 23 14:27 EDT 2014. Contains 247171 sequences.