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A158359
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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).
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2
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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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)!.
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FORMULA
| sum_{0<=k<=n} |T(n,k)| = A000522(n).
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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.
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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
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CROSSREFS
| Cf. A000522
Sequence in context: A008278 A056858 A137251 * A046716 A202605 A123354
Adjacent sequences: A158356 A158357 A158358 * A158360 A158361 A158362
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KEYWORD
| sign,tabl,easy
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AUTHOR
| Gary W. Adamson & Roger L. Bagula (qntmpkt(AT)yahoo.com), Mar 17 2009
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