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%I #5 Dec 02 2018 23:09:15
%S 1,3,3,5,12,8,5,25,40,20,1,30,102,120,48,-7,7,154,364,336,112,-15,-56,
%T 88,672,1184,896,256,-15,-135,-216,624,2592,3600,2304,576,1,-150,-710,
%U -480,3280,9120,10400,5760,1280,33,11,-946,-2860,176,14432,29920,28864,14080,2816,65,396,-60,-4752,-9288,8448,56384
%N A triangular sequence of coefficients made from a product sum of the Pascal/binomial and the Chebyshev T Polynomials: t(n,m)=-Sum[Binomial[n + 1, k + 1]*CoefficientList[ChebyshevT[k + 1, x], x][[m]], {k, m, n}].
%C Roe sums are:
%C {1, 6, 25, 90, 301, 966, 3025, 9330, 28501, 86526, 261625, 788970}.
%C The resulting coefficients are the Chebyshev orthogonal base
%C for Pascal's triangle.
%F t(n,m)=-Sum[Binomial[n + 1, k + 1]*CoefficientList[ChebyshevT[k + 1, x], x][[m]], {k, m, n}].
%e {1},
%e {3, 3},
%e {5, 12, 8},
%e {5, 25, 40, 20},
%e {1, 30, 102, 120, 48},
%e {-7, 7, 154, 364, 336, 112},
%e {-15, -56, 88, 672, 1184, 896, 256},
%e {-15, -135, -216, 624, 2592, 3600, 2304, 576},
%e {1, -150, -710, -480, 3280, 9120, 10400, 5760, 1280},
%e {33, 11, -946, -2860, 176, 14432, 29920,28864, 14080,2816},
%e {65, 396, -60, -4752, -9288, 8448, 56384, 92928, 77568, 33792, 6144},
%e {65, 845, 2652, -1352, -20072, -23504, 55744, 202176, 276224, 203008, 79872, 13312}
%t Clear[t, n, m, k]; t[n_, m_] := -Sum[ Binomial[n + 1, k + 1]*CoefficientList[ChebyshevT[k + 1, x], x][[ m]], {k, m, n}]; Table[Table[t[n, m], {m, 1, n}], {n, 1, 12}]; Flatten[%]
%K uned,sign
%O 1,2
%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 23 2008
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