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A203999 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{i(j+1-1),j(i+1)-1} (A203998). 3
1, -1, -4, -6, 1, 7, 27, 17, -1, -10, -60, -99, -36, 1, 13, 105, 279, 269, 65, -1, -16, -162, -593, -944, -609, -106, 1, 19, 231, 1077, 2405, 2610, 1218, 161, -1, -22, -312, -1767, -5092, -7865, -6264, -2226, -232, 1, 25, 405, 2699, 9541 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

REFERENCES

(For references regarding interlacing roots, see A202605.)

LINKS

Table of n, a(n) for n=1..48.

EXAMPLE

Top of the array:

1....-1

-4....-6.....1

7.... 27....17...-1

-10...-60...-99...-36...1

MATHEMATICA

f[i_, j_] := Max[i (j + 1) - 1, j (i + 1) - 1];

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

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

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

{n, 1, 12}, {i, 1, n}]]    (* A203998 *)

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

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

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

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

Flatten[%]   (* A203999 *)

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

CROSSREFS

Cf. A203998, A202605.

Sequence in context: A030169 A263180 A239809 * A330823 A199371 A156789

Adjacent sequences:  A203996 A203997 A203998 * A204000 A204001 A204002

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 09 2012

STATUS

approved

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Last modified March 2 19:18 EST 2021. Contains 341756 sequences. (Running on oeis4.)