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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. 10
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. - Klaus Brockhaus, May 29 2009

REFERENCES

G. B. Shabat and I. A. Voevodskii, Drawing curves over number fields, The Grothendieck Festschift, vol. 3, Birkhäuser, 1990, 199-227.

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Gareth Jones and David Singerman, Belyi Functions, Hypermaps and Galois Groups, Bull. London Math. Soc., 28 (1996), 561-590.

Yuri Matiyasevich, Generalized Chebyshev polynomials.

G. B. Shabat and A. Zvonkin, Plane trees and algebraic numbers, Contemporary Math., 1994, vol. 178, 233-275.

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 W. 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 December 4 08:27 EST 2020. Contains 338921 sequences. (Running on oeis4.)