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A196307 The cube of the g.f. equals the g.f. of A196306.

%I A196307 #10 Mar 30 2012 18:37:31
%S A196307 1,1,-1,2,-4,9,-22,55,-142,375,-1009,2753,-7599,21178,-59509,168401,
%T A196307 -479477,1372536,-3947678,11402376,-33059314,96177750,-280671373,
%U A196307 821379083,-2409938978,7087502564,-20889306810,61691675424,-182531101523,541000651928,-1606046079955,4774977156350
%N A196307 The cube of the g.f. equals the g.f. of A196306.
%C A196307 A196306 is defined as the Coefficients in the g.f. C(x), where -1 <= A196306(n) <= 1 for all n>1 with initial terms {1,3}, such that C(x)^(1/3) consists entirely of integer coefficients.
%C A196307 Limit a(n+1)/a(n) = -3.1069369226 1299813830 3346689095 3281527516 0860761416 4775926338 8951561634 ...
%H A196307 Paul D. Hanna, <a href="/A196307/b196307.txt">Table of n, a(n) for n = 0..400</a>
%e A196307 G.f.: A(x) = 1 + x - x^2 + 2*x^3 - 4*x^4 + 9*x^5 - 22*x^6 + 55*x^7 - 142*x^8 + 375*x^9 - 1009*x^10 + 2753*x^11 - 7599*x^12 +...
%e A196307 where
%e A196307 A(x)^3 = 1 + 3*x + x^3 - x^6 - x^9 - x^12 - x^18 + x^21 - x^24 - x^30 - x^33 + x^39 - x^42 - x^45 + x^48 +...+ A196306(n)*x^n +...
%e A196307 A196306 begins: [1,3,0,1,0,0,-1,0,0,-1,0,0,-1,0,0,0,0,0,-1,0,0,1,0,0,-1,...].
%o A196307 (PARI) {a(n)=local(A=1+3*x); if(n==0, 1, for(j=1, n, for(k=-1, 1, t=polcoeff((A+k*x^j+x*O(x^j))^(1/3), j);
%o A196307 if(denominator(t)==1, A=A+k*x^j; break))); polcoeff((A+x*O(x^n))^(1/3), n))}
%Y A196307 Cf. A196306, A196308, A106219 (variant).
%K A196307 sign
%O A196307 0,4
%A A196307 _Paul D. Hanna_, Oct 01 2011