A203989 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of {max(i,j)} (A051125).

1, -1, -2, -3, 1, 3, 11, 6, -1, -4, -23, -35, -10, 1, 5, 39, 98, 85, 15, -1, -6, -59, -207, -308, -175, -21, 1, 7, 83, 374, 795, 798, 322, 28, -1, -8, -111, -611, -1694, -2475, -1806, -546, -36, 1, 9, 143, 930, 3185, 6149, 6633, 3696, 870, 45

Offset: 1

Author

Clark Kimberling, Jan 09 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 for a guide to related sequences.

The characteristic polynomial seems be the recurrence relation given by p(n,x) = -x * p(n-1,x) + n * (-1)^(n-1) * sum_{i=0..n-1} x^i * binomial(2n-i-2,i). - Enrique Pérez Herrero, Jan 29 2013

Examples

			Top of the array:
1... -1
-2... -3.... 1
3.... 11... 6... -1
-4... -23.. -35.. -10...1
5.... 39... 98... 85...15.. -1
		

References

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

Crossrefs

Cf. A051125, A202605.

Cf. A181983, A076756.

Programs

Mathematica

f[i_, j_] := Max[i, j];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6th principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]]  (* A051125 *)
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[%]              (* A203989 *)
TableForm[Table[c[n], {n, 1, 10}]]