|
|
A123948
|
|
Triangle read by rows: first row is 1, and n-th row (n > 0) gives the coefficients in the expansion of the characteristic polynomial of the (n - 1)-th Bernstein basis matrix, horizontal flipped.
|
|
0
|
|
|
1, 1, -1, -1, 1, 1, -2, 3, 3, -1, 9, -15, -22, 7, 1, 96, -184, -314, 139, 19, -1, -2500, 5250, 10575, -5375, -1026, 51, 1, -162000, 369900, 842310, -498171, -111179, 7644, 141, -1, 26471025, -64790985, -164634169, 109325076, 28870212, -2322404, -59193, 393, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,7
|
|
COMMENTS
|
The Bernstein basis matrix of order n - 1 is an n X n matrix whose m-th row represents the coefficients in the expansion of the Bernstein basis polynomial defined as binomial(n, m)*x^m*(1 - x)^(n - m), 0 <= m <= n - 1. For n = 0, we obtain the 0 X 0 matrix. The convention is that the characteristic polynomial of the empty matrix is identically 1 (see [de Boor] and [Johnson et al.]). Row n of the present sequence is obtained by taking the characteristic polynomial of the matrix represented by the polynomials binomial(n, m)*x^(n - m)*(1 - x)^m. The resulting matrix is, in fact, the horizontal flipped version of the Bernstein basis matrix of order n (see example). - Franck Maminirina Ramaharo, Oct 19 2018
|
|
REFERENCES
|
Gengzhe Chang and Thomas W. Sederberg, Over and Over Again, The Mathematical Association of America, 1997, Chap. 30.
|
|
LINKS
|
|
|
EXAMPLE
|
Triangle begins:
1;
1, -1;
-1, 1, 1;
-2, 3, 3, -1;
9, -15, -22, 7, 1;
96, -184, -314, 139, 19, -1;
-2500, 5250, 10575, -5375, -1026, 51, 1;
-162000, 369900, 842310, -498171, -111179, 7644, 141, -1;
...
Let n = 6 (i.e., order 5). The corresponding Bernstein basis matrix has the form
1, -5, 10, -10, 5, -1
0, 5, -20, 30, -20, 5
0, 0, 10, -30, 30, -10
0, 0, 0, 10, -20, 10
0, 0, 0, 0, 5, -5
0, 0, 0, 0, 0, 1.
Flipping this matrix horizontally gives the matrix for the polynomials binomial(5, m)*x^(5 - m)*(1 - x)^m, 0 <= m <= 5,
0, 0, 0, 0, 0, 1
0, 0, 0, 0, 5, -5
0, 0, 0, 10, -20, 10
0, 0, 10, -30, 30, -10
0, 5, -20, 30, -20, 5
1, -5, 10, -10, 5, -1
whose characteristic polynomial is -2500 + 5250*x + 10575*x^2 - 5375*x^3 - 1026*x^4 + 51*x^5 + x^6. (End)
|
|
MATHEMATICA
|
M[n_] := Table[CoefficientList[Binomial[n - 1, n - i - 1]*(1 - x)^i*x^(n - i - 1), x], {i, 0, n - 1}];
Join[{1}, Table[CoefficientList[CharacteristicPolynomial[M[d], x], x], {d, 1, 10}]]//Flatten
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|