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

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, 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, 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

Offset: 0

Author

Paul Barry, Oct 30 2004

Keywords

• easy
• sign

Comments

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}.

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,-1)

Formula

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}.