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Search: id:A053117
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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; listen)
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OFFSET
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0,3
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COMMENT
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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
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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
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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
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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
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{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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CROSSREFS
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Cf. A053118, A049310, A000129, A000027.
Sequence in context: A130125 A137336 A115322 this_sequence A121448 A019094 A134082
Adjacent sequences: A053114 A053115 A053116 this_sequence A053118 A053119 A053120
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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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