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.

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

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Oct 26 2006

Keywords

• tabl
• sign

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

Examples

			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;
      ...
From _Franck Maminirina Ramaharo_, Oct 19 2018: (Start)
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)
		

References

• Gengzhe Chang and Thomas W. Sederberg, Over and Over Again, The Mathematical Association of America, 1997, Chap. 30.

Links

• Carl de Boor, An empty exercise
• John Burkardt, BERNSTEIN_POLYNOMIAL - The Bernstein Polynomials
• Charles R. Johnson and Carlos M. Saiago, Eigenvalues, Multiplicities and Graphs, Cambridge University Press, 2018, p. 8.
• Kenneth I. Joy, On-Line Geometric Modeling Notes
• Wikipedia, Bernstein polynomial

Crossrefs

Cf. A045912, A136678.

Programs

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

Extensions

Edited, new name, offset corrected by Franck Maminirina Ramaharo, Oct 19 2018