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A084930
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Triangle of coefficients of Chebyshev polynomials T_{2n+1} (x).
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3
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1, -3, 4, 5, -20, 16, -7, 56, -112, 64, 9, -120, 432, -576, 256, -11, 220, -1232, 2816, -2816, 1024, 13, -364, 2912, -9984, 16640, -13312, 4096, -15, 560, -6048, 28800, -70400, 92160, -61440, 16384, 17, -816, 11424, -71808, 239360, -452608, 487424, -278528, 65536, -19, 1140, -20064, 160512, -695552
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Comment from Herb Conn, HCR 83, Box 93, Custer, SD 57730, Jan 28 2005: "Letting x = 2 Cos 2A, we have the following trigonometric identities:
"Sin 3A = 3*Sin A - 4*Sin^3 A
"Sin 5A = 5*Sin A - 20*Sin^3 A + 16*Sin^5 A
"Sin 7A = 7*Sin A - 56*Sin^3 A + 112*Sin^5 A - 64*Sin^7 A
"Sin 9A = 9*Sin A - 120*Sin^3 A + 432*Sin^5 A - 576*Sin^7 A + 256*Sin^9 A, etc."
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REFERENCES
| Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990. p. 37, eq.(1.96) and p. 4, eq.(1.10).
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LINKS
| 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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 795.
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| Alternate rows of A008310.
a(n,k)=((-1)^(n-k))*(2^(2*k))*binomial(n+1+k,2*k+1)*(2*n+1)/(n+1+k) if n>=k>=0 else 0.
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EXAMPLE
| First few polynomials T_{2n+1}(x) are
1*x
4*x^3 - 3*x
16*x^5 - 20*x^3 + 5*x
64*x^7 - 112*x^5 + 56*x^3 - 7*x
256*x^9 - 576*x^7 + 432*x^5 - 120*x^3 + 9*x
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CROSSREFS
| Cf. A002315, A028297.
Cf. A053120 (coefficient triangle of T-polynomials), A127674 (even indexed T polynomials).
Cf. A127675 (row reversed triangle with different signs).
Sequence in context: A059184 A161961 A161474 * A114336 A048086 A048005
Adjacent sequences: A084927 A084928 A084929 * A084931 A084932 A084933
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KEYWORD
| sign,tabl,easy
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 12 2003
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EXTENSIONS
| More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jun 26 2003
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