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A174986 Numerator coefficients of an infinite sum polynomial:p(x,n)=Sum[(((1 + Sqrt[5])^k - (1 - Sqrt[5])^k)/(2^k* Sqrt[5]))^n*x^k, {k, 0, Infinity}] 0

%I #2 Mar 30 2012 17:34:40

%S -2,-1,1,5,-10,-5,-2,8,8,-2,-10,70,160,-70,-10,-1,12,53,-53,-12,1,20,

%T -400,-3320,6360,3320,-400,-20,-2,66,984,-3568,-3568,984,66,-2,-10,

%U 540,14130,-92880,-178400,92880,14130,-540,-10,-4,352,15840,-184932,-654016

%N Numerator coefficients of an infinite sum polynomial:p(x,n)=Sum[(((1 + Sqrt[5])^k - (1 - Sqrt[5])^k)/(2^k* Sqrt[5]))^n*x^k, {k, 0, Infinity}]

%C Row sums are:

%C {-2, 0, -10, 12, 140, 0, 5560, -5040, -150160, 0,...}.

%C The result seems to be a beta integer version of a Eulerian infinite sum.

%F p(x,n)=Sum[(((1 + Sqrt[5])^k - (1 - Sqrt[5])^k)/(2^k* Sqrt[5]))^n*x^k, {k, 0, Infinity}] out_n,m=Numerator_coefficients(p(x,n)/x)/2^(1 + Floor[n/2])

%e {-2},

%e {-1, 1},

%e {5, -10, -5},

%e {-2, 8, 8, -2},

%e {-10, 70, 160, -70, -10},

%e {-1, 12, 53, -53, -12, 1},

%e {20, -400, -3320, 6360, 3320, -400, -20},

%e {-2, 66, 984, -3568, -3568, 984, 66, -2},

%e {-10, 540, 14130, -92880, -178400, 92880, 14130, -540, -10},

%e {-4, 352, 15840, -184932, -654016, 654016, 184932, -15840, -352, 4}

%t p[x_, n_] = Sum[(((1 + Sqrt[5])^k - (1 - Sqrt[5])^k)/(2^k*Sqrt[5]))^n*x^k, {k, 0, Infinity}];

%t Flatten[Table[CoefficientList[FullSimplify[Numerator[p[x, n]]/x], x]/2^(1 + Floor[n/2]), {n, 1, 10}]]

%K sign,tabl,uned

%O 1,1

%A _Roger L. Bagula_, Apr 03 2010

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