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A099916 Expansion of (1+x^2)^2/(1-x^3+x^6). 2

%I #7 Jun 13 2015 00:51:34

%S 1,0,2,1,1,2,0,1,0,-1,0,-2,-1,-1,-2,0,-1,0,1,0,2,1,1,2,0,1,0,-1,0,-2,

%T -1,-1,-2,0,-1,0,1,0,2,1,1,2,0,1,0,-1,0,-2,-1,-1,-2,0,-1,0,1,0,2,1,1,

%U 2,0,1,0,-1,0,-2,-1,-1,-2,0,-1,0,1,0,2,1,1,2,0,1,0,-1,0,-2,-1,-1,-2,0,-1,0,1,0,2,1,1,2,0,1,0,-1,0

%N Expansion of (1+x^2)^2/(1-x^3+x^6).

%C The denominator is the 18th cyclotomic polynomial. The g.f. is a Chebyshev transform of that of A052931, by the Chebyshev mapping g(x)->(1/(1+x^2))g(x/(1+x^2)). The reciprocal of the 18th cyclotomic polynomial A014027 is given by sum{k=0..n, A099916(n-k)(k/2+1)(-1)^(k/2)(1+(-1)^k)/2}.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,0,0,-1)

%F a(n)=sum{k=0..floor(n/2), C(n-k, k)(-1)^k*sum{j=0..n-2k, C(j, n-2k-2j)3^(3j-n+2k)}}; a(n)=sum{k=0..n, A014027(n-k)C(2, k/2)(1+(-1)^k)/2}.

%K easy,sign

%O 0,3

%A _Paul Barry_, Oct 30 2004

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