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

0,7

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

Table of n, a(n) for n=0..44.

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

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;
      ...
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)

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

Cf. A045912, A136678.

Sequence in context: A144149 A097005 A068008 * A329430 A188886 A131012

Adjacent sequences: A123945 A123946 A123947 * A123949 A123950 A123951

Keyword

tabl,sign

Author

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

Extensions

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

Status

approved