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 A028297 Triangle of coefficients in expansion of cos(n*x) in descending powers of cos(x). 12

%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 Triangle of coefficients in expansion of cos(n*x) in descending powers of cos(x).

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

%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 = A034839 * A007318 [From 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

%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>

%F 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]

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

%F 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

%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 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 - Philippe Deléham, Dec 16 2011

%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]] (* From Jean-François Alcover, May 6 2011 *)

%Y Cf. A028298.

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

%Y Cf. A053120 (table including zeros).

%Y Cf. A118800.

%Y A034839 [From Gary W. Adamson, Nov 28 2008]

%Y Cf. A081277, A124182

%K tabf,easy,sign

%O 0,3

%A _N. J. A. Sloane_.

%E More terms from _David W. Wilson_

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