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Denominator coefficients of infinite over the Fibonacci sequence: p(x,n)=(1 - x)*Sum[Fibonacci[k]^n*x^k, {k, 0, Infinity}]; t(n,m)=Coefficients(Denominator(p(x,n)).
1

%I #4 Jul 05 2012 11:37:37

%S 1,-1,1,1,1,-2,-2,1,1,-3,-6,3,1,1,-4,-19,-4,1,-1,8,40,-60,-40,8,1,1,

%T -13,-104,260,260,-104,-13,1,1,-21,-273,1092,1820,-1092,-273,21,1,1,

%U -33,-747,3894,16270,3894,-747,-33,1,-1,55,1870,-19635,-85085,136136,85085

%N Denominator coefficients of infinite over the Fibonacci sequence: p(x,n)=(1 - x)*Sum[Fibonacci[k]^n*x^k, {k, 0, Infinity}]; t(n,m)=Coefficients(Denominator(p(x,n)).

%C Row sums are:

%C {1, 1, -2, -4, -25, -44, 288, 1276, 22500, 96976, -1707552,...}.

%C The denominator and numerator polynomials appear to be new.

%F p(x,n)=(1 - x)*Sum[Fibonacci[k]^n*x^k, {k, 0, Infinity}]; t(n,m)=Coefficients(Denominator(p(x,n)).

%e {1},

%e {-1, 1, 1},

%e {1, -2, -2, 1},

%e {1, -3, -6, 3, 1},

%e {1, -4, -19, -4, 1},

%e {-1, 8, 40, -60, -40, 8, 1},

%e {1, -13, -104, 260, 260, -104, -13, 1},

%e {1, -21, -273, 1092, 1820, -1092, -273, 21, 1},

%e {1, -33, -747, 3894, 16270, 3894, -747, -33, 1},

%e {-1, 55, 1870, -19635, -85085, 136136, 85085, -19635, -1870, 55, 1},

%e {1, -89, -4895, 83215, 582505, -1514513, -1514513, 582505, 83215, -4895, -89, 1}

%t Clear[t0, p, x, n, m];

%t p[x_, n_] = (1 - x)*Sum[Fibonacci[k]^n*x^k, {k, 0, Infinity}]

%t Table[Denominator[FullSimplify[ExpandAll[p[x, n]]]], {n, 0, 10}];

%t Flatten[%]

%Y A000045, A055870

%K sign,tabl,uned

%O 0,6

%A _Roger L. Bagula_, Feb 04 2009