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Search: id:A039991
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| A039991 |
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Triangle of coefficients of cos(x)^n in polynomial for cos(nx). |
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12
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| 1, 1, 0, 2, 0, -1, 4, 0, -3, 0, 8, 0, -8, 0, 1, 16, 0, -20, 0, 5, 0, 32, 0, -48, 0, 18, 0, -1, 64, 0, -112, 0, 56, 0, -7, 0, 128, 0, -256, 0, 160, 0, -32, 0, 1, 256, 0, -576, 0, 432, 0, -120, 0, 9, 0, 512, 0, -1280, 0, 1120, 0, -400, 0, 50, 0, -1, 1024, 0, -2816, 0, 2816, 0, -1232, 0, 220
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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Also triangle of coefficients of Chebyshev polynomials of first kind (T(n,x)) in decreasing order of powers of x.
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
E. A. Guilleman, Synthesis of Passive Networks, Wiley, 1957, p. 593.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..100
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].
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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a(n, m) = 0 if n<m or m odd, (-1)^{m/2} if m=n is even, ((-1)^(3*m/2))*(2^(n-m-1))*n*binomial(n-1-m/2, n-1-m)/(n-m) else. a(n, m) = 2*a(n-1, m)-a(n-2, m-2), n >= 2, m >= 0; a(n, -2) := 0=: a(n, -1), a(0, 0)=1=a(1, 0).
G.f. for m-th column: 0 if m odd, (1-x)/(1-2*x) if m=0, else ((-1)^(m/2))*(x^m)*(1-x)/(1-2*x)^(m/2+1). For g.f. for row polynomials and row sums, see A053120.
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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.
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CROSSREFS
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Cf. A028297. Essentially same as A008310.
Triangle without zeros: A028297. Without signs: A081265.
Sequence in context: A121298 A121462 A131487 this_sequence A081265 A108643 A133838
Adjacent sequences: A039988 A039989 A039990 this_sequence A039992 A039993 A039994
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KEYWORD
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tabl,easy,sign,nice
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AUTHOR
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Dave Wilson
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EXTENSIONS
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Entry improved by comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jan 11 2000.
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