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A123951 A polynomial of matrices is used to make a triangular sequence. The upper triangular antidiagonal Steinbach matrices are summed over their characteristic polynomial triangular sequences to give a new sequence of matrices: the characteristic polynomials of these new summed matrices are, then, used to make up this triangular sequence.

%I A123951 #13 Dec 10 2016 03:02:51
%S A123951 1,1,-1,-1,-1,1,-1,-3,4,-1,37,-88,69,-19,1,10879,-14344,6831,-1375,99,
%T A123951 -1,-4322473,-40529664,-17486038,3188841,-40896,-2346,1,
%U A123951 -11384127259974047,-783824545942228,1058675233347,505084925760,-64007100,-32568519,23164,-1,-121986767767877481129923
%N A123951 A polynomial of matrices is used to make a triangular sequence. The upper triangular antidiagonal Steinbach matrices are summed over their characteristic polynomial triangular sequences to give a new sequence of matrices: the characteristic polynomials of these new summed matrices are, then, used to make up this triangular sequence.
%C A123951 Basically everything is done twice. The determinants get very large very fast for these matrices: Table[Det[w[[d]]], {d, 1, Length[w]}] {1, -1, -1, 37, 10879, -4322473, -11384127259974047, -121986767767877481129923, -323621163456130064854374309178100414058036559, 189651898964129252384795657180434913387386019400002936829101989683}
%H A123951 P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31
%F A123951 p(n,x) = CharacteristicPolynomial(a(i,j)) p(n,x)->t(n,m) b(i,j) = Sum[t(i,j).a(j,k).{j,1,m}] p'(n,x) = CharacteristicPolynomial(b(i,j)) p'(n,x)->t'(n,m).
%e A123951 {1},
%e A123951 {1, -1},
%e A123951 {-1, -1, 1},
%e A123951 {-1, -3, 4, -1},
%e A123951 {37, -88, 69, -19,1},
%e A123951 {10879, -14344, 6831, -1375, 99, -1},
%e A123951 {-4322473, -40529664, -17486038, 3188841, -40896, -2346, 1}
%t A123951 An[d_] := Table[If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d],x], x], {d, 1, 20}]]; w = Join[{{{1}}}, Table[Sum[MatrixPower[a[[n]][[m + 1]]*An[n], m - 1], {m, 0, Length[a[[n]]] - 1}], {n, 2, 10}]]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[w[[d]], x], x], {d, 1, Length[w]}]]; Flatten[%]
%Y A123951 Cf. A122765, A122771.
%K A123951 uned,tabl,sign
%O A123951 1,8
%A A123951 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 26 2006