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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(NumberatorNumerator(p(x,n)).

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editing

_Roger L. Bagula (rlbagulatftn(AT)yahoo.com), _, Feb 04 2009

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(Numberator(p(x,n)).

1, 0, -1, 1, 0, 1, -2, 1, 0, 1, -3, 1, 1, 0, 1, -4, -4, 1, 0, -1, 8, 9, -23, 6, 1, 0, 1, -13, -41, 106, -41, -13, 1, 0, 1, -21, -146, 484, -152, -186, 19, 1, 0, 1, -33, -492, 1784, 1784, -492, -33, 1, 0, -1, 55, 1359, -10701, -8552, 27128, -7875, -1467, 53, 1, 0, 1, -89

0,7

Row sums are:

{1, 0, 0, 0, -6, 0, 0, 0, 2520, 0, 0,...}.

The denominator and numerator polynomials appear to be new.

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

{1},

{0, -1, 1},

{0, 1, -2, 1},

{0, 1, -3, 1, 1},

{0, 1, -4, -4, 1},

{0, -1, 8, 9, -23, 6, 1},

{0, 1, -13, -41, 106, -41, -13, 1},

{0, 1, -21, -146, 484, -152, -186, 19, 1},

{0, 1, -33, -492, 1784, 1784, -492, -33, 1},

{0, -1, 55, 1359, -10701, -8552, 27128, -7875, -1467, 53, 1},

{0, 1, -89, -3872, 50193, 117271, -327008, 117271, 50193, -3872, -89, 1}

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

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

Table[Numerator[FullSimplify[ExpandAll[p[x, n]]]], {n, 0, 10}];

Table[CoefficientList[Numerator[FullSimplify[ExpandAll[p[x, n]]]], x], {n, 0, 10}];

Flatten[%]

A000045

sign,tabl,uned,new

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 04 2009

approved