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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.

%I A123948 #36 May 26 2019 04:40:45
%S A123948 1,1,-1,-1,1,1,-2,3,3,-1,9,-15,-22,7,1,96,-184,-314,139,19,-1,-2500,
%T A123948 5250,10575,-5375,-1026,51,1,-162000,369900,842310,-498171,-111179,
%U A123948 7644,141,-1,26471025,-64790985,-164634169,109325076,28870212,-2322404,-59193,393,1
%N 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.
%C A123948 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
%D A123948 Gengzhe Chang and Thomas W. Sederberg, Over and Over Again, The Mathematical Association of America, 1997, Chap. 30.
%H A123948 Carl de Boor, <a href="ftp://ftp.cs.wisc.edu/Approx/empty.pdf">An empty exercise</a>
%H A123948 John Burkardt, <a href="https://people.sc.fsu.edu/~jburkardt/m_src/bernstein_polynomial/bernstein_polynomial.html">BERNSTEIN_POLYNOMIAL - The Bernstein Polynomials</a>
%H A123948 Charles R. Johnson and Carlos M. Saiago, <a href="https://doi.org/10.1017/9781316155158">Eigenvalues, Multiplicities and Graphs</a>, Cambridge University Press, 2018, p. 8.
%H A123948 Kenneth I. Joy, <a href="http://graphics.cs.ucdavis.edu/education/CAGDNotes/Bernstein-Polynomials/Bernstein-Polynomials.html">On-Line Geometric Modeling Notes</a>
%H A123948 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bernstein_polynomial">Bernstein polynomial</a>
%e A123948 Triangle begins:
%e A123948         1;
%e A123948         1,     -1;
%e A123948        -1,      1,      1;
%e A123948        -2,      3,      3,      -1;
%e A123948         9,    -15,    -22,       7,       1;
%e A123948        96,   -184,   -314,     139,      19,   -1;
%e A123948     -2500,   5250,  10575,   -5375,   -1026,   51,   1;
%e A123948   -162000, 369900, 842310, -498171, -111179, 7644, 141, -1;
%e A123948       ...
%e A123948 From _Franck Maminirina Ramaharo_, Oct 19 2018: (Start)
%e A123948 Let n = 6 (i.e., order 5). The corresponding Bernstein basis matrix has the form
%e A123948    1, -5,  10, -10,   5,  -1
%e A123948    0,  5, -20,  30, -20,   5
%e A123948    0,  0,  10, -30,  30, -10
%e A123948    0,  0,   0,  10, -20,  10
%e A123948    0,  0,   0,   0,   5,  -5
%e A123948    0,  0,   0,   0,   0,   1.
%e A123948 Flipping this matrix horizontally gives the matrix for the polynomials binomial(5, m)*x^(5 - m)*(1 - x)^m, 0 <= m <= 5,
%e A123948    0,  0,   0,   0,   0,   1
%e A123948    0,  0,   0,   0,   5,  -5
%e A123948    0,  0,   0,  10, -20,  10
%e A123948    0,  0,  10, -30,  30, -10
%e A123948    0,  5, -20,  30, -20,   5
%e A123948    1, -5,  10, -10,   5,  -1
%e A123948 whose characteristic polynomial is -2500 + 5250*x + 10575*x^2 - 5375*x^3 - 1026*x^4 + 51*x^5 + x^6. (End)
%t A123948 M[n_] := Table[CoefficientList[Binomial[n - 1, n - i - 1]*(1 - x)^i*x^(n - i - 1), x], {i, 0, n - 1}];
%t A123948 Join[{1}, Table[CoefficientList[CharacteristicPolynomial[M[d], x], x], {d, 1, 10}]]//Flatten
%Y A123948 Cf. A045912, A136678.
%K A123948 tabl,sign
%O A123948 0,7
%A A123948 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 26 2006
%E A123948 Edited, new name, offset corrected by _Franck Maminirina Ramaharo_, Oct 19 2018