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A203906 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203905.

%I A203906 #18 Sep 20 2019 07:24:49
%S A203906 1,-1,1,-2,1,1,-4,4,-1,1,-6,11,-6,1,1,-8,22,-24,9,-1,1,-10,37,-62,46,
%T A203906 -12,1,1,-12,56,-128,148,-80,16,-1,1,-14,79,-230,367,-314,130,-20,1,1,
%U A203906 -16,106,-376,771,-920,610,-200,25,-1,1,-18,137
%N A203906 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203905.
%C A203906 Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix.  The zeros of p(n) are positive, and they interlace the zeros of p(n+1).  See A202605 for a guide to related sequences.
%C A203906 If we omit the main diagonal of this array and ignore the signs of the entries then the resulting array, reading the rows in reverse order, appears to equal the Riordan array (1/((1 + x)*(1 - x)^3), x/(1 - x)^2), whose generating function begins 1 + (2 + t)*x + (4 + 4*t + t^2)*x^2 + (6 + 11*t + 6*t^2 + t^3)*x^3 + (9 + 24*t + 22*t^2 + 8*t^3 + t^4)*x^4 + .... - _Peter Bala_, Sep 17 2019
%D A203906 (For references regarding interlacing roots, see A202605.)
%e A203906 Top of the array:
%e A203906 1...-1
%e A203906 1...-2....1
%e A203906 1...-4....4...-1
%e A203906 1...-6...11...-6....1
%e A203906 1...-8...22...-24...9...-1
%t A203906 t = {1, 0}; t1 = Flatten[{t, t, t, t, t, t, t, t, t, t}];
%t A203906 f[k_] := t1[[k]];
%t A203906 U[n_] := NestList[Most[Prepend[#, 0]] &, #,
%t A203906 Length[#] - 1] &[Table[f[k], {k, 1, n}]];
%t A203906 L[n_] := Transpose[U[n]];
%t A203906 p[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A203906 c[n_] := CoefficientList[p[n], x]
%t A203906 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A203906 Table[c[n], {n, 1, 12}]
%t A203906 Flatten[%]                         (* A203906 *)
%t A203906 TableForm[Table[c[n], {n, 1, 10}]]
%t A203906 Table[p[n] /. x -> -1, {n, 1, 16}] (* A166516 *)
%Y A203906 Cf. A202605, A166516.
%K A203906 tabf,sign
%O A203906 1,4
%A A203906 _Clark Kimberling_, Jan 08 2012