A084204 -id:A084204 - cp's OEIS frontend

A084203 G.f. A(x) defined by: A(x)^3 consists entirely of integer coefficients between 1 and 3 (A083953); A(x) is the unique power series solution with A(0)=1.

1, 1, 0, 0, 1, -1, 2, -2, 2, 0, -4, 12, -24, 38, -46, 33, 29, -176, 443, -827, 1222, -1310, 433, 2488, -8814, 19528, -33599, 44928, -37805, -17916, 168049, -463252, 921694, -1446018, 1679053, -808620, -2598482, 10515127, -24690122, 44515322, -62719429, 58496244, 10670109, -213311788, 632128236

Offset: 0

Paul D. Hanna, May 19 2003

sign

Does limit_{n ->infinity} a(n)/a(n+1) exist?

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.

Cf. A083953, A084202, A084204-A084212.

kmax = 45;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^3 + O[x]^kmax , x];
r = {};
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 3, a[k-1], Integers] // ToRules]; coes = coes /. r, {k, 1, kmax}];
Array[a, kmax, 0] /. r (* Jean-François Alcover, Jul 26 2018 *)

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.

Original entry on oeis.org

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

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) -> r = -0.2512525316047635 where A(r)=0.

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.

Cf. A083945, A084202-A084204, A084206-A084212.

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 *)

cp's OEIS Frontend

A084203 G.f. A(x) defined by: A(x)^3 consists entirely of integer coefficients between 1 and 3 (A083953); A(x) is the unique power series solution with A(0)=1.

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