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A028297
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Triangle of coefficients in expansion of cos(n*x) in descending powers of cos(x).
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12
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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
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OFFSET
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0,3
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COMMENTS
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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 [From 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
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REFERENCES
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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]
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LINKS
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Alois P. Heinz, Rows n = 0..200, flattened
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FORMULA
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cos(n*x) = 2*cos((n-1)*x)*cos(x) - cos((n-2)*x) (from CRC's Multiple-angle relations). [From 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
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EXAMPLE
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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
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MATHEMATICA
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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]] (* From Jean-François Alcover, May 6 2011 *)
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CROSSREFS
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Cf. A028298.
Reflection of A008310, the main entry. With zeros: A039991.
Cf. A053120 (table including zeros).
Cf. A118800.
A034839 [From Gary W. Adamson, Nov 28 2008]
Cf. A081277, A124182
Sequence in context: A100818 A005291 A106624 * A207537 A114438 A181882
Adjacent sequences: A028294 A028295 A028296 * A028298 A028299 A028300
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KEYWORD
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tabf,easy,sign
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from David W. Wilson
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STATUS
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approved
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