A158359 Triangle T(n,k) read by rows: coefficient [x^(n-k)] of the characteristic polynomial of the n X n matrix A(r,c)=1 (if c > r) and A(r,c)=c (if c <= r).

1, 1, -1, 1, -3, 1, 1, -6, 7, -2, 1, -10, 25, -23, 6, 1, -15, 65, -123, 98, -24, 1, -21, 140, -448, 713, -514, 120, 1, -28, 266, -1288, 3401, -4792, 3204, -720, 1, -36, 462, -3150, 12417, -28599, 36748, -23148, 5040, 1, -45, 750, -6846, 37617, -127935, 265540, -317132, 190224, -40320, 1, -55

Offset: 0

Author

Gary W. Adamson and Roger L. Bagula, Mar 17 2009

Keywords

• sign
• tabl
• easy

Comments

The building matrices A(r,c) have the form [[1,1,1,...],[1,2,1,1,1...],[1,2,3,1,1,...],...,[1,2,3,4,...,n]].

Their determinant and the product of the roots of their characteristic polynomial is (n-1)!.

Examples

			First few characteristic polynomials are:
1;
x - 1;
x^2 - 3x + 1;
x^3 - 6x^2 + 7x - 2;
x^4 - 10x^3 + 25x^2 - 23x + 6;
x^5 - 15x^4 + 65x^3 - 123x^2 + 98x - 24;
x^6 - 21x^5 + 140x^4 - 448x^3 + 713x^2 - 514x + 120;
x^7 - 28x^6 + 266x^5 - 1288x^4 + 3401x^3 - 4792x^2 + 3204x - 720;
x^8 - 36x^7 + 462x^6 - 3150x^5 + 12417x^4 - 28599x^3 + 36748x^2 - 23148x + 5040;
x^9 - 45x^8 + 750x^7 - 6846x^6 + 37617x^5 - 127935x^4 + 265540x^3 - 317132x^2 + 190224x - 40320;
x^10 - 55x^9 + 1155x^8 - 13596x^7 + 99231x^6 - 466488x^5 + 1416955x^4 - 2706992x^3 + 3044412x^2 - 1752336x + 362880
...
Example: 3x3 matrix = [1,1,1; 1,2,1; 1,2,3]; charpoly = x^3 - 6x^2 + 7x - 2,
determinant = 2.
		

Crossrefs

Cf. A000522.

Programs

Maple

A158359 := proc(n,k)
        A := Matrix(1..n,1..n) ;
        for r from 1 to n do
        for c from 1 to n do
                if c > r then
                        A[r,c] := 1 ;
                else
                        A[r,c] := c;
                end if;
        end do;
        end do;
        LinearAlgebra[CharacteristicPolynomial](A,x) ;
        coeftayl(%,x=0,n-k) ;
end proc:
seq(seq(A158359(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Nov 05 2011

Formula

Sum_{k=0..n} |T(n,k)| = A000522(n).