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A099917 Expansion of (1+x^2)^2/(1+x^3+x^6). 1

%I #6 Oct 01 2023 20:13:07

%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 9th cyclotomic polynomial. The g.f. is a Chebyshev transform of that of (-1)^n*A052931(n) by the Chebyshev mapping g(x)->(1/(1+x^2))g(x/(1+x^2)). The reciprocal of the 9th cyclotomic polynomial A014018 is given by sum{k=0..n, A099917(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^k(-1/3)^(n-2k)}}; a(n)=sum{k=0..n, A014018(n-k)C(2, k/2)(1+(-1)^k)/2}.

%Y Cf. A099916.

%K easy,sign

%O 0,3

%A _Paul Barry_, Oct 30 2004

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Last modified December 10 00:12 EST 2023. Contains 367696 sequences. (Running on oeis4.)