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A204126 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(i if i=j and 1 otherwise) (A204125).

%I A204126 #7 Jul 12 2012 00:39:58
%S A204126 1,-1,1,-3,1,2,-8,6,-1,6,-28,29,-10,1,24,-124,155,-75,15,-1,120,-668,
%T A204126 949,-565,160,-21,1,720,-4248,6636,-4564,1610,-301,28,-1,5040,-31176,
%U A204126 52464,-40208,16569,-3892,518,-36,1,40320,-259488,463956
%N A204126 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(i if i=j and 1 otherwise) (A204125).
%C A204126 Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix.  The zeros of p(n) are real, and they interlace the zeros of p(n+1).  See A202605 and A204016 for guides to related sequences.
%D A204126 (For references regarding interlacing roots, see A202605.)
%e A204126 Top of the array:
%e A204126 1....-1
%e A204126 1....-3.....1
%e A204126 2....-8.....6....-1
%e A204126 6....-28....29...-10...1
%t A204126 f[i_, j_] := 1; f[i_, i_] := i;
%t A204126 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204126 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204126 Flatten[Table[f[i, n + 1 - i],
%t A204126   {n, 1, 15}, {i, 1, n}]]  (* A204125 *)
%t A204126 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204126 c[n_] := CoefficientList[p[n], x]
%t A204126 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204126 Table[c[n], {n, 1, 12}]
%t A204126 Flatten[%]                 (* A204126 *)
%t A204126 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204126 Cf. A204125, A202605, A204016.
%K A204126 tabl,sign
%O A204126 1,4
%A A204126 _Clark Kimberling_, Jan 11 2012