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A204170 Array read by rows: row n lists the coefficients of the characteristic polynomial of the n-th principal submatrix of (i*j), as in A003991.

%I A204170 #11 Feb 10 2023 11:56:06
%S A204170 1,-1,0,-5,1,0,0,14,-1,0,0,0,-30,1,0,0,0,0,55,-1,0,0,0,0,0,-91,1,0,0,
%T A204170 0,0,0,0,140,-1,0,0,0,0,0,0,0,-204,1,0,0,0,0,0,0,0,0,285,-1,0,0,0,0,0,
%U A204170 0,0,0,0,-385,1,0,0,0,0,0,0,0,0,0,0,506,-1
%N A204170 Array read by rows:  row n lists the coefficients of the characteristic polynomial of the n-th principal submatrix of (i*j), as in A003991.
%C A204170 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.
%C A204170 p(n,x) = x^n + (-1)^n*s(n)*x^n - 1, where s=A000330 (square pyramidal numbers).
%D A204170 (For references regarding interlacing roots, see A202605.)
%e A204170 Top of the array:
%e A204170   1, -1;
%e A204170   0, -5,  1;
%e A204170   0,  0, 14,  -1;
%e A204170   0,  0,  0, -30, 1;
%t A204170 f[i_, j_] := i*j;
%t A204170 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204170 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204170 Flatten[Table[f[i, n + 1 - i],
%t A204170   {n, 1, 15}, {i, 1, n}]]  (* A003991 *)
%t A204170 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204170 c[n_] := CoefficientList[p[n], x]
%t A204170 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204170 Table[c[n], {n, 1, 12}]
%t A204170 Flatten[%]                (* A204170 *)
%t A204170 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204170 Cf. A003991, A202605, A204016.
%K A204170 tabf,sign
%O A204170 1,4
%A A204170 _Clark Kimberling_, Jan 12 2012