A123588 Triangle read by rows: T(n, k) is the coefficient of x^k in the polynomial 1 - ChT(n, x^(1/2))^2, where ChT(n, x) is the n-th Chebyshev polynomial of the first kind, evaluated at x (0 <= k <= n).

0, 1, -1, 0, 4, -4, 1, -9, 24, -16, 0, 16, -80, 128, -64, 1, -25, 200, -560, 640, -256, 0, 36, -420, 1792, -3456, 3072, -1024, 1, -49, 784, -4704, 13440, -19712, 14336, -4096, 0, 64, -1344, 10752, -42240, 90112, -106496, 65536, -16384, 1, -81, 2160, -22176, 114048, -329472, 559104, -552960, 294912

Offset: 0

Author

Gary W. Adamson and Roger L. Bagula, Nov 12 2006

Keywords

• sign
• tabl

Examples

			Polynomials:
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
Triangle starts:
  0;
  1,  -1;
  0,   4,  -4;
  1,  -9,  24,  -16;
  0,  16, -80,  128, -64;
  1, -25, 200, -560, 640, -256;
		

References

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

Crossrefs

Cf. A000004 (row sums vanish), A114619 (alternating row sums).

Programs

Maple

with(orthopoly): for n from 0 to 9 do seq(coeff(expand((1-T(n,sqrt(x))^2)),x,k), k=0..n) od; # yields sequence in triangular form

Mathematica

row[0] = {0}; row[n_] := CoefficientList[1 - ChebyshevT[n, x^(1/2)]^2, x]; Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Jan 29 2016 *)
T[n_,k_]:=If[k==0,Mod[n,2],(-1)^(n+k-1)*4^(k-1)*(2*Binomial[n+k,2*k]-Binomial[n+k-1,2*k-1])];Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Aug 11 2024 *)

Formula

T(n, 0) = (n mod 2); T(n,k) = (-1)^(n + k - 1)*4^(k - 1)*(2*binomial(n + k, 2*k) - binomial(n + k - 1, 2*k - 1)) for k > 0. - Detlef Meya, Aug 11 2024

Extensions

Edited by N. J. A. Sloane, Dec 03 2006