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

%I A359319 #31 Dec 31 2022 12:46:16
%S A359319 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,
%T A359319 1646,2751,4863,8260,13909,24870,41671,73936,131257,228204,411128,
%U A359319 737620,1292651,2324494,4253857,7487549,13710736,25291179,44938191,82814603
%N A359319 Maximal coefficient of (1 + x) * (1 + x^8) * (1 + x^27) * ... * (1 + x^(n^3)).
%C A359319 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
%H A359319 Vaclav Kotesovec, <a href="/A359319/b359319.txt">Table of n, a(n) for n = 0..250</a>
%F A359319 Conjecture: a(n) ~ sqrt(14) * 2^n / (sqrt(Pi) * n^(7/2)). - _Vaclav Kotesovec_, Dec 30 2022
%t A359319 Table[Max[CoefficientList[Product[1+x^(k^3),{k,n}],x]],{n,0,44}] (* _Stefano Spezia_, Dec 25 2022 *)
%t A359319 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 *)
%o A359319 (PARI) a(n) = vecmax(Vec(prod(i=1, n, (1+x^(i^3))))); \\ _Michel Marcus_, Dec 27 2022
%Y A359319 Cf. A000537, A000578, A001405, A025591, A160235, A279329, A359320.
%K A359319 nonn
%O A359319 0,7
%A A359319 _Ilya Gutkovskiy_, Dec 25 2022