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A204182 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,1)=f(1,j)=1, f(i,i)=2i-1; f(i,j)=0 otherwise; as in A204181. 2
1, -1, 2, -4, 1, 7, -21, 9, -1, 34, -146, 83, -16, 1, 201, -1277, 878, -226, 25, -1, 1266, -13504, 10729, -3340, 500, -36, 1, 6063, -167689, 149971, -53679, 9805, -967, 49, -1, -44190, -2392326, 2368995, -946036, 199829 (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 and A204016 for guides to related sequences.

LINKS

Table of n, a(n) for n=1..40.

EXAMPLE

(For references regarding interlacing roots, see A202605.)

Top of the array:

1....-1

2....-4.....1

7....-21....9....-1

34...-146...83...-16...1

MATHEMATICA

f[i_, j_] := 0; f[1, j_] := 1;

f[i_, 1] := 1; f[i_, i_] := 2 i - 1;

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

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

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

  {n, 1, 15}, {i, 1, n}]]  (* A204181 *)

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[%]                 (* A204182 *)

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

CROSSREFS

Cf. A204181, A202605, A204016.

Sequence in context: A121531 A127554 A182319 * A103324 A221073 A181266

Adjacent sequences:  A204179 A204180 A204181 * A204183 A204184 A204185

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 12 2012

STATUS

approved

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Last modified October 17 22:48 EDT 2018. Contains 316297 sequences. (Running on oeis4.)