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Search: id:A028297
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| A028297 |
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Triangle of coefficients in expansion of cos nx in descending powers of cos x. |
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+0
7
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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
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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 (qntmpkt(AT)yahoo.com), Dec 16 2007
Unsigned = A034839 * A007318 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 28 2008]
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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.
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FORMULA
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Chebyshev coefficients of the first kind for T1 = x, T2 = 2x^2 - 1, ...; are 2^(n-1), -n2^(n-3), [2^(n-5)][n(n-3)]/2!, - [2^(n-7)][n(n-4)(n-5)]/3!, [2^(n-9)][n(n-5)(n-6)(n-7)]/ 4!... - Herb Conn, HCR 83, Box 93, Custer, SD 57730 and Gary W. Adamson (qntmpkt(AT)yahoo.com), May 28 2003
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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.
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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 (qntmpkt(AT)yahoo.com), Nov 28 2008]
Sequence in context: A100818 A005291 A106624 this_sequence A114438 A109195 A032662
Adjacent sequences: A028294 A028295 A028296 this_sequence 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 (njas(AT)research.att.com).
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EXTENSIONS
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More terms from David W. Wilson (davidwwilson(AT)comcast.net)
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