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A123583 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. 8
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; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

All row sum are zero. Row sums of absolute values are in A114619. [From Klaus Brockhaus, May 29 2009]

REFERENCES

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.

LINKS

Table of n, a(n) for n=0..79.

Yuri Matiyasevich, Generalized Chebyshev polynomials.

EXAMPLE

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.

MATHEMATICA

w = Table[CoefficientList[1 - ChebyshevT[n, x]^2, x], {n, 0, 10}]; Flatten[w]

PROG

(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

CROSSREFS

Cf. A123588.

Sequence in context: A112919 A019201 A137660 * A236112 A226787 A140574

Adjacent sequences:  A123580 A123581 A123582 * A123584 A123585 A123586

KEYWORD

tabf,sign

AUTHOR

Gary Adamson and Roger L. Bagula, Nov 12 2006

EXTENSIONS

Edited by N. J. A. Sloane, Mar 09 2008

STATUS

approved

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Last modified July 23 23:07 EDT 2014. Contains 244873 sequences.