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%I #14 Feb 01 2015 16:41:11
%S 1,1,-1,-1,-1,1,1,0,1,-1,-1,0,0,-1,1,1,0,0,0,1,-1,-1,0,0,0,0,-1,1,1,0,
%T 0,0,0,0,1,-1,-1,0,0,0,0,0,0,-1,1,1,0,0,0,0,0,0,0,1,-1,-1,0,0,0,0,0,0,
%U 0,0,-1,1,1,0,0,0,0,0,0,0
%N Triangle read by rows where the n-th row gives the coefficients of the characteristic polynomial for a Fibonacci-type matrix with a=1 and b=1.
%C Row sums are 1, 0, -1, 1, -1, 1, -1, 1, -1, 1, ... .
%H J. Cigler, <a href="http://www.fq.math.ca/Scanned/41-1/cigler.pdf">q-Fibonacci polynomials</a>, Fibonacci Quarterly 41 (2003) 31-40.
%F The n-th row contains the coefficients (from lowest-order to highest-order) of the characteristic polynomial of the matrix with (i,j)-entry given by: if(i = j = n, 1, if(j = n and i = 1, 1, if(i = j + 1, 1, 0))).
%F For n >= 2, the n-th row of the triangle consists of (-1)^(n+1), followed by n-2 zeros, followed by (-1)^(n+1) and (-1)^n. - _Nathaniel Johnston_, Apr 27 2011
%e Triangle begins:
%e 1,
%e 1, -1,
%e -1, -1, 1,
%e 1, 0, 1, -1,
%e -1, 0, 0, -1, 1,
%e 1, 0, 0, 0, 1, -1,
%e -1, 0, 0, 0, 0, -1, 1,
%e 1, 0, 0, 0, 0, 0, 1, -1,
%e -1, 0, 0, 0, 0, 0, 0, -1, 1,
%e 1, 0, 0, 0, 0, 0, 0, 0, 1, -1,
%e -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1,
%e ...
%e For n = 4, the matrix is {{0, 0, 0, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 1}}.
%t T[n_, m_, d_] := If[ n == m == d, 1, If[m == d && n == 1, 1, If[n == m + 1, 1, 0]]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; Table[Det[M[d]], {d, 1, 10}]; Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]; a = Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]],x], {d, 1, 10}]]; Flatten[a]
%Y The triangle when reversed is very similar to A141679. - _N. J. A. Sloane_, Dec 14 2014
%K tabl,easy,sign
%O 0,1
%A _Roger L. Bagula_, Apr 06 2008
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