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%I #11 May 10 2019 22:22:21
%S 1,1,-1,1,-3,1,1,-6,7,-2,1,-10,25,-23,6,1,-15,65,-123,98,-24,1,-21,
%T 140,-448,713,-514,120,1,-28,266,-1288,3401,-4792,3204,-720,1,-36,462,
%U -3150,12417,-28599,36748,-23148,5040,1,-45,750,-6846,37617,-127935,265540,-317132,190224,-40320,1,-55
%N 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).
%C 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]].
%C Their determinant and the product of the roots of their characteristic polynomial is (n-1)!.
%F Sum_{k=0..n} |T(n,k)| = A000522(n).
%e First few characteristic polynomials are:
%e 1;
%e x - 1;
%e x^2 - 3x + 1;
%e x^3 - 6x^2 + 7x - 2;
%e x^4 - 10x^3 + 25x^2 - 23x + 6;
%e x^5 - 15x^4 + 65x^3 - 123x^2 + 98x - 24;
%e x^6 - 21x^5 + 140x^4 - 448x^3 + 713x^2 - 514x + 120;
%e x^7 - 28x^6 + 266x^5 - 1288x^4 + 3401x^3 - 4792x^2 + 3204x - 720;
%e x^8 - 36x^7 + 462x^6 - 3150x^5 + 12417x^4 - 28599x^3 + 36748x^2 - 23148x + 5040;
%e x^9 - 45x^8 + 750x^7 - 6846x^6 + 37617x^5 - 127935x^4 + 265540x^3 - 317132x^2 + 190224x - 40320;
%e x^10 - 55x^9 + 1155x^8 - 13596x^7 + 99231x^6 - 466488x^5 + 1416955x^4 - 2706992x^3 + 3044412x^2 - 1752336x + 362880
%e ...
%e Example: 3x3 matrix = [1,1,1; 1,2,1; 1,2,3]; charpoly = x^3 - 6x^2 + 7x - 2,
%e determinant = 2.
%p A158359 := proc(n,k)
%p A := Matrix(1..n,1..n) ;
%p for r from 1 to n do
%p for c from 1 to n do
%p if c > r then
%p A[r,c] := 1 ;
%p else
%p A[r,c] := c;
%p end if;
%p end do;
%p end do;
%p LinearAlgebra[CharacteristicPolynomial](A,x) ;
%p coeftayl(%,x=0,n-k) ;
%p end proc:
%p seq(seq(A158359(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Nov 05 2011
%Y Cf. A000522.
%K sign,tabl,easy
%O 0,5
%A _Gary W. Adamson_ and _Roger L. Bagula_, Mar 17 2009
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