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

%I A228161 #23 Feb 24 2025 04:37:07
%S A228161 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,
%T A228161 7,780,1189,496,99,12,1,1,8,2911,6930,3905,980,143,14,1,0,9,10864,
%U A228161 40391,30744,9701,1704,195,16,1,-1,10,40545,235416,242047,96030,20305,2716,255,18,1
%N A228161 Number triangle associated to Chebyshev polynomials of the second kind.
%C A228161 Compare the definition of U_n(x) with the definition of the Dirichlet kernel.
%C A228161 U_n(x) is defined as sin((n+1)*arccos(x))/sin(arccos(x)).
%C A228161 U_n(x) is a polynomial in x with integer coefficients for all n >=0.
%C A228161 The initial term is U_0(0).
%C A228161 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),....
%H A228161 T. D. Noe, <a href="/A228161/b228161.txt">Rows n = 0..100 of triangle, flattened</a>
%H A228161 Wikipedia, <a href="http://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>
%F A228161 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.
%e A228161 Triangle begins:
%e A228161   1,
%e A228161   0, 1,
%e A228161  -1, 2,  1,
%e A228161   0, 3,  4,  1,
%e A228161   1, 4, 15,  6, 1,
%e A228161   0, 5, 56, 35, 8, 1,
%e A228161   ...
%t A228161 nn = 10; Flatten[Table[ChebyshevU[i - j, j], {i, 0, nn}, {j, 0, i}]] (* _T. D. Noe_, Aug 16 2013 *)
%Y A228161 Cf. A101124 (number triangle for Chebyshev polynomials of the first kind).
%Y A228161 Cf. A133156 (coefficients of powers of x in U_n(x)).
%K A228161 sign,easy,tabl
%O A228161 0,5
%A A228161 _Jonny Griffiths_, Aug 14 2013
%E A228161 More terms from _Michel Marcus_, Feb 24 2025