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A163810 Expansion of (1 - x) * (1 - x^2) * (1 - x^3) / (1 - x^6) in powers of x.

%I A163810 #16 Apr 18 2022 10:35:07
%S A163810 1,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,
%T A163810 1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,
%U A163810 1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1
%N A163810 Expansion of (1 - x) * (1 - x^2) * (1 - x^3) / (1 - x^6) in powers of x.
%H A163810 G. C. Greubel, <a href="/A163810/b163810.txt">Table of n, a(n) for n = 0..1000</a>
%H A163810 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1).
%F A163810 Euler transform of length 6 sequence [ -1, -1, -1, 0, 0, 1].
%F A163810 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 2 * u * (1 - u) * (2 - v) - (v - u^2).
%F A163810 a(3*n) = 0 unless n=0. a(6*n + 1) = a(6*n + 2) = -1, a(6*n + 4) = a(6*n + 5) = a(0) = 1.
%F A163810 a(-n) = -a(n) unless n=0. a(n+3) = -a(n) unless n=0 or n=-3.
%F A163810 G.f.: (1 - x)^2 / (1 - x + x^2).
%e A163810 G.f. = 1 - x - x^2 + x^4 + x^5 - x^7 - x^8 + x^10 + x^11 - x^13 - x^14 + ...
%t A163810 Join[{1},LinearRecurrence[{1, -1},{-1, -1},104]] (* _Ray Chandler_, Sep 15 2015 *)
%o A163810 (PARI) {a(n) = (n==0) + [0, -1, -1, 0, 1, 1][n%6 + 1]};
%o A163810 (PARI) {a(n) = (n==0) + (-1)^n * kronecker(-3, n)};
%Y A163810 A163806(n) = -a(n) unless n=0. A106510(n) = (-1)^n * a(n).
%Y A163810 Convolution inverse of A028310. Series reversion of A109081.
%K A163810 sign,easy
%O A163810 0,1
%A A163810 _Michael Somos_, Nov 07 2007