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A196344 Coefficients of g.f. A(x) where -2 <= a(n) <= 2 for all n>1, with initial terms {1,5}, such that A(x)^(1/5) consists entirely of integer coefficients.

%I A196344 #9 Mar 13 2015 22:55:24
%S A196344 1,5,0,0,0,1,0,0,0,0,-2,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,
%T A196344 0,0,1,0,0,0,0,-2,0,0,0,0,-1,0,0,0,0,-2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,
%U A196344 0,0,0,0,-1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,2,0,0,0,0,-1,0,0,0,0,2,0,0,0,0,-1,0,0,0,0,2,0,0,0,0,-1,0,0,0,0,2,0,0,0,0,-1,0,0,0,0,2,0,0,0,0,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0
%N A196344 Coefficients of g.f. A(x) where -2 <= a(n) <= 2 for all n>1, with initial terms {1,5}, such that A(x)^(1/5) consists entirely of integer coefficients.
%C A196344 A(z) = 0 at z = -0.1999360614 0060333411 6048445601 7524456163 6606200034 1004259693 ...
%H A196344 Paul D. Hanna, <a href="/A196344/b196344.txt">Table of n, a(n) for n = 0..1000</a>
%e A196344 G.f.: A(x) = 1 + 5*x + x^5 - 2*x^10 + x^15 - x^20 + x^35 - 2*x^40 - x^45 - 2*x^50 + x^55 + 2*x^65 - x^70 + 2*x^80 +...
%e A196344 where the following series consists entirely of integer coefficients:
%e A196344 A(x)^(1/5) = 1 + x - 2*x^2 + 6*x^3 - 21*x^4 + 80*x^5 - 320*x^6 + 1326*x^7 - 5637*x^8 + 24434*x^9 - 107542*x^10 +...+ A196345(n)*x^n +...
%o A196344 (PARI) {a(n)=local(A=1+5*x); if(n==0, 1, if(n%5==0,for(j=1, n, for(k=-2, 2, t=polcoeff((A+k*x^j+x*O(x^j))^(1/5), j);
%o A196344 if(denominator(t)==1, A=A+k*x^j; break)))); polcoeff(A+x*O(x^n), n))}
%Y A196344 Cf. A196345 (5th root), A196346 (quintisection), A106222 (variant).
%K A196344 sign
%O A196344 0,2
%A A196344 _Paul D. Hanna_, Oct 01 2011