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A164117 Expansion of (1 - x) * (1 - x^10) / ((1 - x^2) * (1 - x^4) * (1 - x^5)) in powers of x.

%I A164117 #15 Sep 08 2022 08:45:47
%S A164117 1,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,
%T A164117 -1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,
%U A164117 1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2,-1,1,-1,2
%N A164117 Expansion of (1 - x) * (1 - x^10) / ((1 - x^2) * (1 - x^4) * (1 - x^5)) in powers of x.
%C A164117 Convolution inverse of A164118.
%H A164117 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 1).
%F A164117 Euler transform of length 10 sequence [-1, 1, 0, 1, 1, 0, 0, 0, 0, -1].
%F A164117 a(n) = -b(n) where b(n) is multiplicative with b(2) = -1, b(2^e) = -2 if e>1, b(p^e) = 1 if p>2.
%F A164117 a(n) = a(-n) for all n in Z. a(n+4) = a(n) unless n=0 or n=-4.
%F A164117 G.f.: (1 - x + x^2 - x^3 + x^4) / (1 - x^4).
%F A164117 a(n) = (-1)^n * A164415(n).
%e A164117 G.f. = 1 - x + x^2 - x^3 + 2*x^4 - x^5 + x^6 - x^7 + 2*x^8 - x^9 + x^10 + ...
%t A164117 CoefficientList[Series[(1-x)(1-x^10)/((1-x^2)(1-x^4)(1-x^5)),{x,0,120}], x] (* _Harvey P. Dale_, Nov 28 2014 *)
%o A164117 (PARI) {a(n) = (-1)^n - (n==0) + (n%4==0)};
%o A164117 (PARI) {a(n) = -(n==0) + [2, -1, 1, -1][n%4 + 1]};
%o A164117 (Magma) m:=100; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)*(1-x^10)/((1-x^2)*(1-x^4)*(1-x^5)))); // _G. C. Greubel_, Sep 25 2018
%Y A164117 Cf. A164415, A164118.
%K A164117 sign,easy
%O A164117 0,5
%A A164117 _Michael Somos_, Aug 10 2009