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%I #4 Dec 10 2016 16:47:00
%S 1,1,0,1,0,1,1,-16,0,0,1,0,-24,0,1,1,-128,0,-256,0,0,1,0,-288,0,-200,
%T 0,1,1,-1024,0,-24576,0,-1296,0,0,1,0,-3456,0,-12000,0,-784,0,1,1,
%U -8192,0,-2621440,0,-590976,0,-4096,0,0
%N Antidiagonal expansion of rational polynomial with factors: p(x,n) = If[n == 0, 1/(1 - x), x*ChebyshevU[n, x]/ChebyshevT[n + 1, x]].
%C Row sums are {1, 1, 2, -15, -22, -383, -486, -26895, -16238, -3224703, ...}.
%C The rational function here is associated with tan(n*arctan(x)).
%F p(x,n) = If[n == 0, 1/(1 - x), x*ChebyshevU[n, x]/ChebyshevT[n + 1, x]];
%F a(n,m) = (n+1)^m*expansion(p(x,n));
%F t(n,m) = antidiagonal(t(n,m)).
%e {1},
%e {1, 0},
%e {1, 0, 1},
%e {1, -16, 0, 0},
%e {1, 0, -24, 0, 1},
%e {1, -128, 0, -256, 0, 0},
%e {1, 0, -288, 0, -200, 0, 1},
%e {1, -1024, 0, -24576, 0, -1296, 0, 0},
%e {1, 0, -3456, 0, -12000, 0, -784, 0, 1},
%e {1, -8192, 0, -2621440, 0, -590976, 0, -4096, 0, 0}
%t p[x_, n_] = If[n == 0, 1/(1 - x), x*ChebyshevU[n, x]/ChebyshevT[n + 1, x]];
%t a = Table[Table[(n + 1)^(m + 1)*SeriesCoefficient[Series[p[x, n], {x, 0, 50}], m], { m, 0, 20}], {n, 0, 20}];
%t Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
%t Flatten[%]
%K sign,tabl,uned
%O 0,8
%A _Roger L. Bagula_, Feb 15 2010
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