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Search: id:A123583
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| A123583 |
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Triangle read by rows: T(n, k) is the coefficient of x^k in the polynomial 1 - T_n(x)^2, where T_n(x) is the n-th Chebyshev polynomial of the first kind. |
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9
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| 0, 1, 0, -1, 0, 0, 4, 0, -4, 1, 0, -9, 0, 24, 0, -16, 0, 0, 16, 0, -80, 0, 128, 0, -64, 1, 0, -25, 0, 200, 0, -560, 0, 640, 0, -256, 0, 0, 36, 0, -420, 0, 1792, 0, -3456, 0, 3072, 0, -1024, 1, 0, -49, 0, 784, 0, -4704, 0, 13440, 0, -19712, 0, 14336, 0, -4096, 0, 0, 64, 0, -1344, 0, 10752, 0, -42240, 0, 90112, 0, -106496, 0, 65536, 0
(list; graph; listen)
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OFFSET
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0,7
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COMMENT
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All row sum are zero. Row sums of absolute values are in A114619. [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), May 29 2009]
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REFERENCES
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Gareth Jones and David Singerman, Belyi Functions, Hypermaps and Galois Groups, Bull. London Math. Soc., 28 (1996), 561-590.
G. B. Shabat and I. A. Voevodskii, Drawing curves over number fields, The Grothendieck Festschift, vol. 3, Birkhaeuser, 1990, 199-227.
G. B. Shabat and A. Zvonkin, Plane trees and algebraic numbers, Contemporary Math., 1994, vol. 178, 233-275.
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LINKS
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Yuri Matiyasevich, Generalized Chebyshev polynomials.
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EXAMPLE
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First few rows of the triangle are:
[ 0 ]
[ 1, 0, -1 ]
[ 0, 0, 4, 0, -4 ]
[ 1, 0, -9, 0, 24, 0, -16 ]
[ 0, 0, 16, 0, -80, 0, 128, 0, -64 ]
[ 1, 0, -25, 0, 200, 0, -560, 0, 640, 0, -256 ]
[ 0, 0, 36, 0, -420, 0, 1792, 0, -3456, 0, 3072, 0, -1024 ]
First few polynomials are:
0,
1 - x^2,
4 x^2 - 4 x^4,
1 - 9 x^2 + 24 x^4 - 16 x^6,
16 x^2 - 80 x^4 + 128 x^6 - 64 x^8,
1 - 25 x^2 + 200 x^4 - 560 x^6 + 640 x^8 - 256 x^10,
36 x^2 - 420 x^4 + 1792 x^6 - 3456 x^8 + 3072 x^10 - 1024 x^12.
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MATHEMATICA
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w = Table[CoefficientList[1 - ChebyshevT[n, x]^2, x], {n, 0, 10}]; Flatten[w]
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PROGRAM
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(MAGMA) [0] cat &cat[ Coefficients(1-ChebyshevT(n)^2): n in [1..8] ];
(PARI) v=[]; for(n=0, 8, v=concat(v, vector(2*n+1, j, polcoeff(1-poltchebi(n)^2, j-1)))); v
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CROSSREFS
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Cf. A123588.
Sequence in context: A112919 A019201 A137660 this_sequence A140574 A010636 A152977
Adjacent sequences: A123580 A123581 A123582 this_sequence A123584 A123585 A123586
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KEYWORD
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tabf,sign
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AUTHOR
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Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Nov 12 2006
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EXTENSIONS
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Edited by N. J. A. Sloane (njas(AT)research.att.com), Mar 09 2008
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