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 A203991 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of {(i+j)*min(i,j)} (A203990). 3

%I #7 Jul 12 2012 00:39:54

%S 2,-1,7,-10,1,38,-71,28,-1,281,-610,357,-60,1,2634,-6329,4620,-1253,

%T 110,-1,29919,-77530,65613,-23348,3514,-182,1,399342,-1098271,1036044,

%U -442349,90800,-8442,280,-1,6125265

%N Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of {(i+j)*min(i,j)} (A203990).

%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 for a guide to related sequences.

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

%e Top of the array:

%e 2.... -1

%e 7.... -10... 1

%e 38... -71... 28... -1

%e 281.. -610.. 357.. -60... 1

%t f[i_, j_] := (i + j) Min[i, j];

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

%t TableForm[m[6]] (* 6x6 principal submatrix *)

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

%t {n, 1, 12}, {i, 1, n}]] (* A203990 *)

%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[%] (* A203991 *)

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

%Y Cf. A203990, A202605.

%K tabl,sign

%O 1,1

%A _Clark Kimberling_, Jan 09 2012

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Last modified June 23 06:39 EDT 2024. Contains 373629 sequences. (Running on oeis4.)