%I #10 Aug 26 2023 21:47:43

%S 1,1,1,0,1,2,-1,-1,2,4,0,-3,-4,4,8,1,1,-8,-12,8,16,0,5,6,-20,-32,16,

%T 32,-1,-1,18,24,-48,-80,32,64,0,-7,-8,56,80,-112,-192,64,128,1,1,-32,

%U -40,160,240,-256,-448,128,256,0,9,10,-120,-160,432,672,-576,-1024,256,512,-1,-1,50,60,-400,-560,1120,1792,-1280,-2304,512,1024

%N A triangle of coefficients formed by adding Chebyshev T(x,n) to x*U(x,n): p(x,n)=ChebyshevT[n, x] + x*ChebyshevU[n, x].

%C Row sums are: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

%C The idea of joining the cosine like Chebyshev to the sine like in this manner is to get rid of the most of the zeros.

%C A {1} was added to the top of the triangle to make it complete with row sums starting at 1.

%F p(x,n)=ChebyshevT[n, x] + x*ChebyshevU[n, x]; t(n,m)=Coefficients(p(x,n)).

%e {1},

%e {1, 1},

%e {0, 1, 2},

%e {-1, -1, 2, 4},

%e {0, -3, -4, 4, 8},

%e {1, 1, -8, -12, 8, 16},

%e {0, 5, 6, -20, -32, 16, 32},

%e {-1, -1, 18, 24, -48, -80,32, 64},

%e {0, -7, -8, 56, 80, -112, -192, 64, 128},

%e {1, 1, -32, -40, 160, 240, -256, -448, 128, 256},

%e {0, 9, 10, -120, -160, 432, 672, -576, -1024, 256, 512},

%e {-1, -1, 50, 60, -400, -560, 1120, 1792, -1280, -2304, 512, 1024}

%t Clear[p, x, n, a]; p[x_, n_] = ChebyshevT[n, x] + x*ChebyshevU[n, x]; Table[FullSimplify[Expand[p[x, n]]], {n, 0, 10}]; Join[{{1}}, Table[CoefficientList[FullSimplify[Expand[p[x, n]]], x], {n, 0, 10}]]; Flatten[%]

%Y Cf. A053120.

%K uned,sign,tabl,less

%O 0,6

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 25 2008