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A228161
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Number triangle associated to Chebyshev polynomials of the second kind.
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4
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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
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OFFSET
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0,5
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COMMENTS
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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),....
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LINKS
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FORMULA
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The polynomials can be computed with U_{n+1}(x) = 2xU_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) = 2x.
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MATHEMATICA
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nn = 10; Flatten[Table[ChebyshevU[i - j, j], {i, 0, nn}, {j, 0, i}]] (* T. D. Noe, Aug 16 2013 *)
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CROSSREFS
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Cf. A101124 (number triangle for Chebyshev polynomials of the first kind).
Cf. A133156 (coefficients of powers of x in U_n(x)).
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KEYWORD
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AUTHOR
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STATUS
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approved
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