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A084205 G.f. A(x) defined by: A(x)^5 consists entirely of integer coefficients between 1 and 5 (A083945); A(x) is the unique power series solution with A(0)=1.

%I A084205 #10 Jul 26 2018 18:00:27
%S A084205 1,1,-1,3,-8,24,-76,252,-854,2950,-10343,36706,-131570,475576,
%T A084205 -1731357,6342042,-23356185,86421603,-321111661,1197586539,
%U A084205 -4481348585,16819759474,-63302097780,238835017492,-903165412289,3422512973645,-12994514592311,49425252955926
%N A084205 G.f. A(x) defined by: A(x)^5 consists entirely of integer coefficients between 1 and 5 (A083945); A(x) is the unique power series solution with A(0)=1.
%C A084205 Limit a(n)/a(n+1) -> r = -0.2512525316047635 where A(r)=0.
%H A084205 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006.
%H A084205 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://doi.org/10.1016/j.jcta.2006.03.018">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%t A084205 kmax = 30;
%t A084205 A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
%t A084205 coes = CoefficientList[A[x]^5 + O[x]^(kmax + 1), x];
%t A084205 r = {a[0] -> 1, a[1] -> 1};
%t A084205 coes = coes /. r;
%t A084205 Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 5, a[k-1], Integers] // ToRules];
%t A084205 coes = coes /. r, {k, 3, kmax+1}];
%t A084205 Table[a[k], {k, 0, kmax}] /. r (* _Jean-François Alcover_, Jul 26 2018 *)
%Y A084205 Cf. A083945, A084202-A084204, A084206-A084212.
%K A084205 sign
%O A084205 0,4
%A A084205 _Paul D. Hanna_, May 20 2003