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A122185 Cube of the Steinbach matrix as a characteristic polynomial triangular array.

%I A122185 #13 Sep 21 2017 03:23:38
%S A122185 1,1,-1,-1,-4,1,-1,4,11,-1,1,4,-21,-23,1,1,-4,-31,60,42,-1,-1,-4,41,
%T A122185 100,-171,-69,1,-1,4,51,-140,-400,381,106,-1,1,4,-61,-180,729,1060,
%U A122185 -823,-154,1,1,-4,-71,220,1158,-2136,-3032,1561,215,-1,-1,-4,81,260,-1687,-3612,7721,6887,-2874,-290,1,-1,4,91,-300,-2316
%N A122185 Cube of the Steinbach matrix as a characteristic polynomial triangular array.
%C A122185 After finding the relationship of A[i,j]^2=Min[i,j] (A122073) and I[n]-A[i,j] (A122160), A[i,j]^3 suggested itself. Matrices: {{3, 2}, {2, 1}}, {{6, 5, 3}, {5, 4,2}, {3, 2, 1}}, {{10, 9, 7, 4}, {9, 8, 6, 3}, {7, 6, 4, 2}, {4, 3, 2,1}}, {{15, 14, 12, 9, 5}, {14, 13, 11, 8, 4}, {12, 11, 9,6, 3}, {9, 8, 6, 4, 2}, {5, 4, 3, 2, 1}}
%H A122185 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 A122185 A(i,j)^3-->P(n,k) P(n,k)->T(n,m)
%e A122185 {1},
%e A122185 {1, -1},
%e A122185 {-1, -4, 1},
%e A122185 {-1, 4, 11, -1},
%e A122185 {1, 4, -21, -23, 1},
%e A122185 {1, -4, -31, 60, 42, -1},
%e A122185 {-1, -4, 41, 100, -171, -69, 1},
%e A122185 {-1, 4, 51, -140, -400, 381, 106, -1}
%t A122185 An[d_] := Table[If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[MatrixPower[An[d], 3], x], x], {d, 1, 20}]]; Flatten[%]
%K A122185 uned,tabl,sign
%O A122185 1,5
%A A122185 _Gary W. Adamson_ and _Roger L. Bagula_, Oct 17 2006