%I #2 Oct 12 2012 14:54:56

%S 1,1,-1,-4,-1,1,-36,9,5,-1,576,64,-61,-5,1,14400,-1600,-1045,105,14,

%T -1,-518400,-32400,29088,1593,-385,-14,1,-25401600,1587600,1079200,

%U -63481,-11521,579,30,-1,1625702400,65028096,-55637056,-2062912,523001,16045

%N A triangle of matrix polynomials: M(d)=Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]]*Transpose[Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]]].

%C Row sums are:

%C {1, 0, -4, -23, 575, 11873, -520517, -22809194, 1633567982, 122963894441,

%C -13226887049109,...}.

%C Example matrix:

%C M(3)={{1, 2, 3},

%C {2, 4, 0},

%C {3, 0, 0}}

%F M(d)=Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]]*Transpose[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 {-4, -1, 1},

%e {-36, 9, 5, -1},

%e {576, 64, -61, -5, 1},

%e {14400, -1600, -1045, 105, 14, -1},

%e {-518400, -32400, 29088, 1593, -385, -14, 1},

%e {-25401600, 1587600, 1079200, -63481, -11521, 579, 30, -1},

%e {1625702400, 65028096, -55637056, -2062912, 523001, 16045, -1563, -30, 1},

%e {131681894400, -5267275776, -3616612416, 138811968, 28130697, -981153, -75500, 2167, 55, -1},

%e {-13168189440000, -365783040000, 301206312000, 7960416400, -2038537300, -48112905, 5258465, 99136, -4851, -55, 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}]]*Transpose[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