OFFSET
0,7
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
FORMULA
Conjecture: a(n) ~ sqrt(14) * 2^n / (sqrt(Pi) * n^(7/2)). - Vaclav Kotesovec, Dec 30 2022
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 *)
PROG
(PARI) a(n) = vecmax(Vec(prod(i=1, n, (1+x^(i^3))))); \\ Michel Marcus, Dec 27 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 25 2022
STATUS
approved