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Search: id:A053120
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| A053120 |
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Triangle of coefficients of Chebyshev's T(n,x) polynomials (powers of x in increasing order). |
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161
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| 1, 0, 1, -1, 0, 2, 0, -3, 0, 4, 1, 0, -8, 0, 8, 0, 5, 0, -20, 0, 16, -1, 0, 18, 0, -48, 0, 32, 0, -7, 0, 56, 0, -112, 0, 64, 1, 0, -32, 0, 160, 0, -256, 0, 128, 0, 9, 0, -120, 0, 432, 0, -576, 0, 256, -1, 0, 50, 0, -400, 0, 1120, 0, -1280, 0, 512, 0, -11, 0, 220, 0, -1232, 0, 2816, 0, -2816
(list; table; graph; listen)
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OFFSET
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0,6
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COMMENT
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a(n,m) = A039991(n,n-m).
G.f. for row polynomials T(n,x) (signed triangle): (1-x*z)/(1-2*x*z+z^2). If unsigned:(1-x*z)/(1-2*x*z-z^2).
Row sums (signed triangle): A000012 (powers of 1). Row sums (unsigned triangle): A001333(n).
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REFERENCES
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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, Rows 0 to 100 of triangle, flattened
W. Lang, Rows n=0..20
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 n+m odd; a(n, m)= (-1)^n/2 if m=0 (n even); else a(n, m)=((-1)^((n+m)/ 2+m))*(2^(m-1))*n*binomial((n+m)/2-1, m-1)/m.
Recursion for n >= 2: a(n, m) = 2*a(n-1, m-1)-a(n-2, m), a(n, m)=0 if n<m, a(n, -1) := 0, a(0, 0)=1=a(1, 1).
G.f. for m-th column (signed triangle): 1/(1+x^2) if m=0 else (2^(m-1))*(x^m)*(1-x^2)/(1+x^2)^(m+1).
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EXAMPLE
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{1}; {0,1}; {-1,0,2}; {0,-3,0,4}; {1,0,-8,0,8};... E.g. fourth row (n=3) corresponds to polynomial T(3,x)= -3*x+4*x^3.
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PROGRAM
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(MAGMA) &cat[ Coefficients(ChebyshevT(n)): n in [0..11] ]; - Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Mar 08 2008
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CROSSREFS
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Cf. A039991, A000012, A001333.
Sequence in context: A073739 A046767 A115720 this_sequence A008743 A029179 A008721
Adjacent sequences: A053117 A053118 A053119 this_sequence A053121 A053122 A053123
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KEYWORD
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easy,nice,sign,tabl
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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