A137405 - OEIS

%I #3 Mar 30 2012 17:34:26

%S 1,3,-1,-4,-10,1,12,44,23,-1,0,576,-864,-288,316,71,-1,-7104,-608,

%T 5800,1168,-670,-108,1,45248,44096,-19712,-21712,116,1768,151,-1,

%U 450432,135424,-346176,-66560,69008,7344,-3204,-204,1,2240512,-5071104,-1422080,2168128,188672,-212928,-10924,5960,265,-1

%N Triangular sequence from coefficients of characteristic polynomial of n X n prime element matrices: M=A.B.A^(-1); (A(3) is singular): examples; A(4)= {{2, 3, 5, 7, 11}, {3, 5, 7, 11, 13}, {5, 7, 11, 13, 17}, {7, 11, 13, 17, 19}, {11, 13, 17, 19, 23}} B(4)= {{3, 5, 7, 11, 13}, {5, 7, 11, 13, 17}, {7, 11, 13, 17, 19}, {11, 13, 17, 19, 23}, {13, 17, 19, 23, 29}}.

%C Row sums are (I put in zero for the singular matrix level): {1, 2, -13, 78, 0, -190, -1521, 49954, 246065, -2113500, 9798535, 74936314, ...}.

%C As far as I know, the only other use of this type of method is in deriving A086515.

%F for n<=m,=d: A(d)=Prime[n + m + 1] B(d)=Prime[n + m + 2] p(x,d)=CharacteristicPolynomial[A.B.A^(-1)] out_n,m=Coefficient(p(x,d))

%e {1},

%e {3, -1},

%e {-4, -10, 1},

%e {12, 44,23, -1},

%e {0},

%e {576, -864, -288, 316, 71, -1},

%e {-7104, -608, 5800, 1168, -670, -108, 1},

%e {45248, 44096, -19712, -21712, 116, 1768,151, -1},

%e {450432, 135424, -346176, -66560, 69008, 7344, -3204, -204, 1},

%e {2240512, -5071104, -1422080, 2168128, 188672, -212928, -10924, 5960, 265, -1},

%e {5292544, -27469312, 33984128, 4845952, -6953248, -363232, 455688, 16064, -9714, -336, 1},

%e {-88076288, 158813184, 142065920, -134528512, -22576128, 18750592, 1438864, -925536, -41344, 15148, 415, -1}

%t a = Table[Prime[n], {n, 1, 500}]; aa[d_] := Table[a[[n + m + 1]], {n, 0, d}, {m, 0, d}]; bb [d_] := Table[a[[n + m + 2]], {n, 0, d}, {m, 0, d}]; M [d_] := aa[d].bb[d].MatrixPower[aa[d], -1] g = Join[{1}, Table[If[d == 3, 0, CharacteristicPolynomial[M[d], x]], {d, 0, 10}]]; a0 = Join[{{1}}, Table[CoefficientList[If[d == 3,0, CharacteristicPolynomial[M[d], x]], x], {d, 0, 10}]]; Flatten[{{1}, {3, -1}, {-4, -10, 1}, {12, 44, 23, -1}, {0}, {576, -864, -288, 316, 71, -1}, {-7104, -608, 5800, 1168, -670, -108, 1}, {45248, 44096, -19712, -21712, 116, 1768, 151, -1}, {450432, 135424, -346176, -66560, 69008, 7344, -3204, -204, 1}, {2240512, -5071104, -1422080, 2168128, 188672, -212928, -10924, 5960, 265, -1}, {5292544, -27469312, 33984128, 4845952, -6953248, -363232, 455688, 16064, -9714, -336, 1}, {-88076288, 158813184, 142065920, -134528512, -22576128, 18750592, 1438864, -925536, -41344, 15148, 415, -1}}]

%Y Cf. A086515.

%K tabf,uned,sign

%O 1,2

%A _Roger L. Bagula_, Apr 14 2008