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

0,5

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.

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

Example

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;

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 *)

Crossrefs

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

Sequence in context: A179399 A371401 A080721 * A373504 A289710 A243594

Adjacent sequences: A123585 A123586 A123587 * A123589 A123590 A123591

Keyword

sign,tabl,changed

Author

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

Extensions

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

Status

approved