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 A137307 A triangular sequence of coefficients of even plus odd Chebyshev polynomials, A053120: q(x,n)=T(x,2*n-1]+T(x,2*n). 0

%I

%S 1,1,-1,1,2,1,-3,-8,4,8,-1,5,18,-20,-48,16,32,1,-7,-32,56,160,-112,

%T -256,64,128,-1,9,50,-120,-400,432,1120,-576,-1280,256,512,1,-11,-72,

%U 220,840,-1232,-3584,2816,6912,-2816,-6144,1024,2048,-1,13,98,-364,-1568,2912,9408,-9984,-26880,16640,39424,-13312,-28672

%N A triangular sequence of coefficients of even plus odd Chebyshev polynomials, A053120: q(x,n)=T(x,2*n-1]+T(x,2*n).

%C The row sums are all 2 and double integrations are all orthogonal except for the zero to one level.

%C This arose from an idea of Chladni Chebyshev's:

%C q(Exp[i*t],n)=T(Cos[2*Pi*t),2*n-1)+T(Sin(2*Pi*t),2*n)

%C which are strange looping spirals.

%F q(x,n)=T(x,2*n-1]+T(x,2*n).

%e {1, 1},

%e {-1, 1, 2},

%e {1, -3, -8, 4, 8},

%e {-1, 5,18, -20, -48, 16, 32},

%e {1, -7, -32, 56, 160, -112, -256, 64, 128},

%e {-1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512},

%e {1, -11, -72, 220, 840, -1232, -3584, 2816, 6912, -2816, -6144, 1024, 2048},

%e {-1, 13, 98, -364, -1568, 2912, 9408, -9984, -26880, 16640, 39424, -13312, -28672, 4096, 8192},

%e {1, -15, -128, 560, 2688, -6048, 21504, 28800, 84480, -70400, -180224, 92160, 212992, -61440, -131072, 16384, 32768},

%e {-1, 17, 162, -816, -4320, 11424, 44352, -71808, -228096, 239360, 658944, -452608, -1118208, 487424, 1105920, -278528, -589824, 65536, 131072},

%e {1, -19, -200, 1140, 6600, -20064, -84480, 160512, 549120, -695552, -2050048, 1770496, 4659200, -2723840, -6553600, 2490368, 5570560, -1245184, -2621440, 262144, 524288}

%t Q[x_, n_] := ChebyshevT[2*n - 1, x] + ChebyshevT[2*n, x]; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a0 = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a0]

%Y Cf. A053120.

%K uned,tabl,sign

%O 1,5

%A _Roger L. Bagula_, Apr 20 2008

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