A204170 - OEIS

%I #11 Feb 10 2023 11:56:06

%S 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 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 0,0,0,0,-385,1,0,0,0,0,0,0,0,0,0,0,506,-1

%N 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 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 p(n,x) = x^n + (-1)^n*s(n)*x^n - 1, where s=A000330 (square pyramidal numbers).

%D (For references regarding interlacing roots, see A202605.)

%e Top of the array:

%e   1, -1;

%e   0, -5,  1;

%e   0,  0, 14,  -1;

%e   0,  0,  0, -30, 1;

%t f[i_, j_] := i*j;

%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

%t TableForm[m[8]] (* 8x8 principal submatrix *)

%t Flatten[Table[f[i, n + 1 - i],

%t   {n, 1, 15}, {i, 1, n}]]  (* A003991 *)

%t p[n_] := CharacteristicPolynomial[m[n], x];

%t c[n_] := CoefficientList[p[n], x]

%t TableForm[Flatten[Table[p[n], {n, 1, 10}]]]

%t Table[c[n], {n, 1, 12}]

%t Flatten[%]                (* A204170 *)

%t TableForm[Table[c[n], {n, 1, 10}]]

%Y Cf. A003991, A202605, A204016.

%K tabf,sign

%O 1,4

%A _Clark Kimberling_, Jan 12 2012