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A203992 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (A143182 in square format). 2
1, -1, -3, -2, 1, 8, 14, 3, -1, -20, -56, -40, -4, 1, 48, 184, 224, 90, 5, -1, -112, -544, -936, -672, -175, -6, 1, 256, 1504, 3344, 3480, 1680, 308, 7, -1, -576, -3968, -10816, -14784, -10560, -3696, -504, -8, 1, 1280, 10112, 32640, 55328 (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

-3... -1.... 1

8.... 14... 3... -1

-20.. -56.. -40.. -4... 1

MATHEMATICA

f[i_, j_] := Max[i - j + 1, j - i + 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}]] (* A143182 in square format *)

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[%] (* A203992 *)

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

CROSSREFS

Cf. A143182, A202605.

Sequence in context: A104552 A210803 A204144 * A204019 A196846 A101413

Adjacent sequences: A203989 A203990 A203991 * A203993 A203994 A203995

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 09 2012

STATUS

approved

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Last modified December 5 02:30 EST 2022. Contains 358572 sequences. (Running on oeis4.)