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A053117
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Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).
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12
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1, 0, 2, -1, 0, 4, 0, -4, 0, 8, 1, 0, -12, 0, 16, 0, 6, 0, -32, 0, 32, -1, 0, 24, 0, -80, 0, 64, 0, -8, 0, 80, 0, -192, 0, 128, 1, 0, -40, 0, 240, 0, -448, 0, 256, 0, 10, 0, -160, 0, 672, 0, -1024, 0, 512, -1, 0, 60, 0, -560, 0, 1792, 0, -2304, 0, 1024, 0, -12, 0, 280, 0, -1792, 0, 4608, 0, -5120, 0, 2048, 1, 0, -84, 0, 1120, 0, -5376, 0, 11520
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n,m)= (2^m)*A049310(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
| Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011; http://repository.wit.ie/1693/1/AoifeThesis.pdf
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
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LINKS
| T. D. Noe, Rows n=0..100 of triangle, flattened
R. Pemantle and M. C. Wilson, Asymptotics of multivariate sequences, I: smooth points of the singular variety
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(n, m) := 0 if n<m or n+m odd, else ((-1)^((n+m)/2+m))*(2^m)*binomial((n+m)/2, m); a(n, m) = -a(n-2, m)+2*a(n-1, m-1), a(n, -1) := 0 =: a(-1, m), a(0, 0)=1, a(n, m)= 0 if n<m or n+m odd; G.f. for m-th column (signed triangle): (1/(1+x^2)^(m+1))*(2*x)^m.
If n and k are of the same parity then a(n,k)=(-1)^((n-k)/2)*sum(binomial((n+k)/2,i)*binomial((n+k)/2-i,(n-k)/2),i=0..k) and a(n,k)=0 otherwise. - Milan R. Janjic (agnus(AT)blic.net), Apr 13 2008
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EXAMPLE
| {1}; {0,2}; {-1,0,4}; {0,-4,0,8}; {1,0,-12,0,16};... E.g. fourth row (n=3) {0,-4,0,8} corresponds to polynomial U(3,x)= -4*x+8*x^3.
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MATHEMATICA
| Flatten[ Table[ CoefficientList[ ChebyshevU[n, x], x], {n, 0, 12}]](* From Jean-François Alcover, Nov 24 2011 *)
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CROSSREFS
| Cf. A053118, A049310, A000129, A000027.
Sequence in context: A130125 A137336 A115322 * A121448 A019094 A134082
Adjacent sequences: A053114 A053115 A053116 * A053118 A053119 A053120
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KEYWORD
| easy,nice,sign,tabl,changed
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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