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A084204 G.f. A(x) defined by: A(x)^4 consists entirely of integer coefficients between 1 and 4 (A083954); A(x) is the unique power series solution with A(0)=1.

%I A084204 #13 Sep 04 2019 23:58:22
%S A084204 1,1,-1,3,-7,20,-58,177,-554,1769,-5739,18866,-62684,210146,-709882,
%T A084204 2413743,-8253995,28366316,-97916761,339326189,-1180068800,4116957243,
%U A084204 -14404398636,50530280752,-177684095927,626181400993,-2211215950469,7823025701314,-27724997048327
%N A084204 G.f. A(x) defined by: A(x)^4 consists entirely of integer coefficients between 1 and 4 (A083954); A(x) is the unique power series solution with A(0)=1.
%C A084204 Limit a(n)/a(n+1) -> r = -0.269562488839799 where A(r)=0.
%H A084204 Robert Israel, <a href="/A084204/b084204.txt">Table of n, a(n) for n = 0..1000</a>
%H A084204 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 A084204 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.
%p A084204 g:= 1: a[0]:= 1:
%p A084204 for n from 1 to 50 do
%p A084204   a[n]:= -floor((coeff(g^4,x,n)-1)/4);
%p A084204   g:= g + a[n]*x^n;
%p A084204 od:
%p A084204 seq(a[n],n=0..50); # _Robert Israel_, Sep 04 2019
%t A084204 kmax = 30;
%t A084204 A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
%t A084204 coes = CoefficientList[A[x]^4 + O[x]^(kmax + 1), x];
%t A084204 r = {a[0] -> 1, a[1] -> 1}; coes = coes /. r;
%t A084204 Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 4, a[k-1], Integers] // ToRules];
%t A084204 coes = coes /. r, {k, 3, kmax + 1}];
%t A084204 Table[a[k], {k, 0, kmax}] /. r (* _Jean-François Alcover_, Jul 26 2018 *)
%Y A084204 Cf. A083954, A084202, A084203, A084205-A084212.
%K A084204 sign
%O A084204 0,4
%A A084204 _Paul D. Hanna_, May 20 2003