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Search: id:A053118
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| A053118 |
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Triangle of coefficients of Chebyshev's U(n,x) polynomials (exponents in decreasing order). |
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4
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| 1, 2, 0, 4, 0, -1, 8, 0, -4, 0, 16, 0, -12, 0, 1, 32, 0, -32, 0, 6, 0, 64, 0, -80, 0, 24, 0, -1, 128, 0, -192, 0, 80, 0, -8, 0, 256, 0, -448, 0, 240, 0, -40, 0, 1, 512, 0, -1024, 0, 672, 0, -160, 0, 10, 0, 1024, 0, -2304, 0, 1792, 0, -560, 0, 60, 0, -1, 2048, 0, -5120, 0, 4608, 0, -1792, 0, 280, 0, -12, 0, 4096, 0, -11264, 0, 11520, 0, -5376
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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a(n,m)= A053117(n,n-m) = 2^(n-m)*A049310(n,n-m).
G.f. for row polynomials U(n,x) (signed triangle): 1/(1-2*x*z+z^2). Unsigned triangle |a(n,m)| has Fibonacci polynomials F(n+1,2*x) as row polynomials with G.f. 1/(1-2*x*z-z^2).
Row sums (unsigned triangle) A000129(n+1) (Pell). Row sums (signed triangle) A000027(n+1) (natural numbers).
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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 n=0..100 of triangle, flattened
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, else ((-1)^(3*m/2))*(2^(n-m))*binomial(n-m/2, n-m); a(n, m) = 2*a(n-1, m) - a(n-2, m-2), a(n, -2) := 0 =: a(n, -1), a(0, 0)=1, a(n, m)= 0 if n<m or m odd; G.f. for m-th column (signed triangle): (-1)^(3*m/2)*x^m/(1-2*x)^(m/2+1) if m >= 0 is even else 0.
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EXAMPLE
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{1}; {2,0}; {4,0,-1}; {8,0,-4,0}; {16,0,-12,0,1};... E.g. fourth row (n=3) {8,0,-4,0} corresponds to polynomial U(3,x)= 8*x^3-4*x.
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CROSSREFS
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Cf. A053117, A049310, A000129, A000027.
Triangle reflected without zeros: A008312 (the main entry).
Sequence in context: A081120 A102392 A051517 this_sequence A119607 A164297 A109578
Adjacent sequences: A053115 A053116 A053117 this_sequence A053119 A053120 A053121
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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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