A228161 Number triangle associated to Chebyshev polynomials of the second kind.

1, 0, 1, -1, 2, 1, 0, 3, 4, 1, 1, 4, 15, 6, 1, 0, 5, 56, 35, 8, 1, -1, 6, 209, 204, 63, 10, 1, 0, 7, 780, 1189, 496, 99, 12, 1, 1, 8, 2911, 6930, 3905, 980, 143, 14, 1, 0, 9, 10864, 40391, 30744, 9701, 1704, 195, 16, 1, -1, 10, 40545, 235416, 242047, 96030, 20305, 2716, 255, 18, 1

Offset: 0

Author

Jonny Griffiths, Aug 14 2013

Keywords

• sign
• easy
• tabl

Comments

Compare the definition of U_n(x) with the definition of the Dirichlet kernel.

U_n(x) is defined as sin((n+1)*arccos(x))/sin(arccos(x)).

U_n(x) is a polynomial in x with integer coefficients for all n >=0.

The initial term is U_0(0).

The triangle is given here as U_0(0), U_1(0), U_1(1), U_2(0), U_2(1), U_2(2), U_3(0),....

Examples

			Triangle begins:
  1,
  0, 1,
 -1, 2,  1,
  0, 3,  4,  1,
  1, 4, 15,  6, 1,
  0, 5, 56, 35, 8, 1,
  ...
		

Links

• T. D. Noe, Rows n = 0..100 of triangle, flattened
• Wikipedia, Chebyshev polynomials

Crossrefs

Cf. A101124 (number triangle for Chebyshev polynomials of the first kind).

Cf. A133156 (coefficients of powers of x in U_n(x)).

Programs

Mathematica

nn = 10; Flatten[Table[ChebyshevU[i - j, j], {i, 0, nn}, {j, 0, i}]] (* T. D. Noe, Aug 16 2013 *)

Formula

The polynomials can be computed with U_{n+1}(x) = 2*x*U_n(x) - U_{n-1}(x), U_{n+1}(x) = ((U_n(x))^2-1)/U_{n-1}(x), where in each case U_0(x) = 1; U_1(x) = 2*x.

Extensions

More terms from Michel Marcus, Feb 24 2025