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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.
3
1, 1, -1, 3, -8, 24, -76, 252, -854, 2950, -10343, 36706, -131570, 475576, -1731357, 6342042, -23356185, 86421603, -321111661, 1197586539, -4481348585, 16819759474, -63302097780, 238835017492, -903165412289, 3422512973645, -12994514592311, 49425252955926
OFFSET
0,4
COMMENTS
Limit a(n)/a(n+1) -> r = -0.2512525316047635 where A(r)=0.
LINKS
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
MATHEMATICA
kmax = 30;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^5 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 5, a[k-1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax+1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, May 20 2003
STATUS
approved