OFFSET
0,5
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)!.
FORMULA
Sum_{k=0..n} |T(n,k)| = A000522(n).
EXAMPLE
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.
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
CROSSREFS
KEYWORD
AUTHOR
Gary W. Adamson and Roger L. Bagula, Mar 17 2009
STATUS
approved