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%I #8 Aug 09 2015 01:24:29
%S 1,1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,-1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,
%T 1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,-1,-1,
%U -1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N Triangle read by rows, formed from the coefficients of characteristic polynomials of the following sequence of matrices: 2 X 2 {{0, 1}, {1, 1}}, 3 X 3 {{0, 1, 0}, {0, 0, 1}, {1, 1, 1}}, 4 X 4 {{0, 1,0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 1, 1, 1}}, 5 X 5 {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {1, 1, 1, 1, 1}}, ...
%C Row sums are {1, 0, -1, 2, -3, 4, -5, 6, -7, 8, -9}.
%F B(x, n) = If[n > 1, (-1)^n*(x^n - Sum[x^m, {m, 0, n - 1}])]
%e Triangular array:
%e {1},
%e {1, -1},
%e {-1, -1, 1},
%e {1, 1, 1, -1},
%e {-1, -1, -1, -1, 1},
%e {1, 1, 1,1, 1, -1},
%e {-1, -1, -1, -1, -1, -1,1},
%e {1, 1, 1, 1, 1, 1, 1, -1},
%e {-1, -1, -1, -1, -1, -1, -1, -1, 1}
%t An[d_] := Table[If[n == d, 1, If[m == n + 1, 1, 0]], {n, 1, d}, {m, 1, d}]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; Flatten[%]
%t Clear[B, x, n] B[x, 0] = 1; B[x, 1] = -x + 1; B[x_, n_] := B[x, n] = If[n > 1, (-1)^n*(x^n - Sum[x^m, {m, 0, n - 1}])]; Table[ExpandAll[B[x, n]], {n, 0, 10}]; a = Table[CoefficientList[B[x, n], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[B[x, n], x]], {n, 0, 10}]
%Y Cf. A000073, A000045.
%K tabl,sign
%O 1,1
%A _Gary W. Adamson_ and _Roger L. Bagula_, Oct 18 2006, Mar 18 2008
%E Edited by _N. J. A. Sloane_, Apr 14 2008
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