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.

1, 1, -2, 7, -27, 115, -510, 2343, -11029, 52896, -257457, 1268098, -6307546, 31633044, -159757597, 811708539, -4145882814, 21273287952, -109603172373, 566748274099, -2940175511195, 15297961574259, -79808998488751, 417373462315834

Offset: 0

Author

Paul D. Hanna, May 20 2003

Keywords

• sign

Comments

Limit a(n)/a(n+1) -> r = -0.1815238859919 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.

Crossrefs

Cf. A083946, A084202-A084205, A084207-A084212.

Programs

Mathematica

kmax = 25;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^6 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 6, 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 *)