login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{3i+j-3,i+3j-3} (A204008).
3

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

%S 1,-1,-11,-6,1,40,70,15,-1,-116,-328,-240,-28,1,304,1176,1456,610,45,

%T -1,-752,-3680,-6408,-4704,-1295,-66,1,1792,10592,23760,25080,12432,

%U 2436,91,-1,-4160,-28800,-79040

%N Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{3i+j-3,i+3j-3} (A204008).

%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 1.....-1

%e -11....-6.....1

%e 40.....70....15....-1

%e -116...-328..-240....1

%t f[i_, j_] := Max[3 i + j - 3, 3 j + i - 3];

%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}]] (* A204008 *)

%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[%] (* A204011 *)

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

%Y Cf. A204008, A202605.

%K tabl,sign

%O 1,3

%A _Clark Kimberling_, Jan 09 2012