A204135 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the Delannoy matrix (A008288).

1, -1, 2, -4, 1, 8, -28, 17, -1, 64, -384, 424, -80, 1, 1024, -10624, 19400, -7700, 401, -1, 32768, -598016, 1748224, -1225536, 161618, -2084, 1, 2097152, -68550656, 319410176, -363159040, 95891872

Offset: 1

Author

Clark Kimberling, Jan 12 2012

Keywords

• tabl
• sign

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.

Examples

			Top of the array:
1....-1
2....-4.....1
8....-28....17....-1
64...-384...424...-80...1
The interlacing of zeros is illustrated by these zeros (truncated):
p(1):  1
p(2): .58, 3.41
p(3): .36, 1.44, 15.19
p(4): .21, .87, 4.53, 74.3
p(5): .12, .59, 2.14, 17.22, 380.91
		

References

• (For references regarding interlacing roots, see A202605.)

Crossrefs

Cf. A008288, A202605, A204016.

Programs

Mathematica

f[i_, 1] := 1; f[1, j_] := 1;
f[i_, j_] := f[i, j - 1] + f[i - 1, j - 1] + f[i - 1, j]
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}]]  (*  Delannoy, A008288 *)
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, 8}]
Flatten[%]                 (* 204135 *)
TableForm[Table[c[n], {n, 1, 6}]]