A359319 Maximal coefficient of (1 + x) * (1 + x^8) * (1 + x^27) * ... * (1 + x^(n^3)).

1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 5, 7, 10, 14, 18, 27, 36, 62, 95, 140, 241, 370, 607, 1014, 1646, 2751, 4863, 8260, 13909, 24870, 41671, 73936, 131257, 228204, 411128, 737620, 1292651, 2324494, 4253857, 7487549, 13710736, 25291179, 44938191, 82814603

Offset: 0

Author

Ilya Gutkovskiy, Dec 25 2022

Keywords

• nonn

Comments

Conjecture: Maximal coefficient of Product_{k=1..n} (1 + x^(n^m)) ~ sqrt(4*m + 2) * 2^n / (sqrt(Pi) * n^(m + 1/2)), for m>=0. - Vaclav Kotesovec, Dec 30 2022

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A000537, A000578, A001405, A025591, A160235, A279329, A359320.

Programs

Mathematica

Table[Max[CoefficientList[Product[1+x^(k^3),{k,n}],x]],{n,0,44}] (* Stefano Spezia, Dec 25 2022 *)
nmax = 100; poly = ConstantArray[0, nmax^2*(nmax + 1)^2/4 + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k^3 + 1]], {j, k^2*(k + 1)^2/4, k^3, -1}]; Print[k, " ", Max[poly]], {k, 2, nmax}]; (* Vaclav Kotesovec, Dec 29 2022 *)

PARI

a(n) = vecmax(Vec(prod(i=1, n, (1+x^(i^3))))); \\ Michel Marcus, Dec 27 2022

Formula

Conjecture: a(n) ~ sqrt(14) * 2^n / (sqrt(Pi) * n^(7/2)). - Vaclav Kotesovec, Dec 30 2022