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%I #2 Oct 12 2012 14:54:56
%S 1,1,-1,-2,-1,1,-6,3,3,-1,24,8,-13,-3,1,120,-40,-61,17,6,-1,-720,-180,
%T 360,75,-45,-6,1,-5040,1260,2368,-529,-309,55,10,-1,40320,8064,-18296,
%U -3256,2545,363,-115,-10,1,362880,-72576,-155736,28728,21609,-3465
%N A triangle of matrix polynomials: M(d)=Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]].
%C Row sums are:
%C {1, 0, -2, -1, 17, 41, -515, -2186, 29616, 180497, -2730775,...}.
%F M(d)=Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]];
%F out_(n,m)=coefficients(characteristicpolynomial(M(n),x),x)
%e {1},
%e {1, -1},
%e {-2, -1, 1},
%e {-6, 3, 3, -1},
%e {24, 8, -13, -3, 1},
%e {120, -40, -61, 17, 6, -1},
%e {-720, -180, 360, 75, -45, -6, 1},
%e {-5040, 1260, 2368, -529, -309, 55, 10, -1},
%e {40320, 8064, -18296, -3256, 2545, 363, -115, -10, 1},
%e { 362880, -72576, -155736, 28728, 21609, -3465, -1092, 135, 15, -1},
%e {-3628800, -604800, 1499160, 229700, -210946, -28077, 12001, 1246, -245, -15, 1}
%t Clear[M, T, d, a, x, a0];
%t M[d_] := Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]];
%t a0 = Table[M[d], {d, 1, 10}];
%t Table[Det[M[d]], {d, 1, 10}];
%t Table[CharacteristicPolynomial[M[d], x], {d, 1, 10}];
%t Flatten[a]
%t Join[{1}, Table[Apply[Plus, CoefficientList[ Expand[CharacteristicPolynomial[M[n], x]], x]], {n, 1, 10}];
%K sign,tabl
%O 0,4
%A _Roger L. Bagula_ and _Gary W. Adamson_, Mar 17 2009
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