A204180 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)= i; f(i,j)=0 otherwise; as in A204179.

1, -1, 1, -3, 1, 1, -9, 6, -1, -2, -32, 32, -10, 1, -34, -132, 183, -81, 15, -1, -324, -604, 1159, -655, 170, -21, 1, -2988, -2860, 8137, -5589, 1825, -316, 28, -1, -28944, -11864, 62852, -51184, 19894, -4326, 539, -36, 1, -300816, -8568

Offset

1,4

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.

References

(For references regarding interlacing roots, see A202605.)

Links

Robert Israel, Table of n, a(n) for n = 1..10010 (rows 1 to 140, flattened)

Formula

From Robert Israel, Jun 26 2018: (Start)

p(n,x) = (1 - Sum_{k=2..n} 1/((1-x)*(k-x)))*Product_{k=1..n} (k - x).

p(n+1,x) = (n+1-x)*p(n,x) - Gamma(n+1-x)/Gamma(2-x). (End)

Example

Top of the array:
   1,  -1;
   1,  -3,   1;
   1,  -9,   6,  -1;
  -2, -32,  32, -10,   1;

Maple

f:= proc(n) local P;
  P:= normal(mul(i-lambda, i=1..n)*(1 - add(1/(lambda-1)/(lambda-i), i=2..n)));
  seq(coeff(P, lambda, i), i=0..n);
end proc:
seq(f(n), n=1..20); # Robert Israel, Jun 26 2018

Mathematica

f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1; f[i_, i_] := i;
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}]]  (* A204179 *)
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[%]                 (* A204180 *)
TableForm[Table[c[n], {n, 1, 10}]]

Crossrefs

Cf. A204179, A202605, A204016.

Sequence in context: A050153 A294950 A294609 * A319729 A106340 A156610

Adjacent sequences: A204177 A204178 A204179 * A204181 A204182 A204183

Keyword

tabf,sign

Author

Clark Kimberling, Jan 12 2012

Status

approved