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

%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