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A084206 G.f. A(x) defined by: A(x)^6 consists entirely of integer coefficients between 1 and 6 (A083946); A(x) is the unique power series solution with A(0)=1.

%I A084206 #10 Jul 26 2018 17:50:05
%S A084206 1,1,-2,7,-27,115,-510,2343,-11029,52896,-257457,1268098,-6307546,
%T A084206 31633044,-159757597,811708539,-4145882814,21273287952,-109603172373,
%U A084206 566748274099,-2940175511195,15297961574259,-79808998488751,417373462315834
%N A084206 G.f. A(x) defined by: A(x)^6 consists entirely of integer coefficients between 1 and 6 (A083946); A(x) is the unique power series solution with A(0)=1.
%C A084206 Limit a(n)/a(n+1) -> r = -0.1815238859919 where A(r)=0.
%H A084206 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 A084206 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 A084206 kmax = 25;
%t A084206 A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
%t A084206 coes = CoefficientList[A[x]^6 + O[x]^(kmax + 1), x];
%t A084206 r = {a[0] -> 1, a[1] -> 1};
%t A084206 coes = coes /. r;
%t A084206 Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 6, a[k-1], Integers] // ToRules];
%t A084206 coes = coes /. r, {k, 3, kmax + 1}];
%t A084206 Table[a[k], {k, 0, kmax}] /. r (* _Jean-François Alcover_, Jul 26 2018 *)
%Y A084206 Cf. A083946, A084202-A084205, A084207-A084212.
%K A084206 sign
%O A084206 0,3
%A A084206 _Paul D. Hanna_, May 20 2003