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a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^3) + 1 + 1/x^(k^3)).
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%I #13 Jan 22 2024 01:50:08

%S 1,1,1,1,1,1,3,3,5,17,31,61,139,309,701,1651,3849,8929,22295,53777,

%T 131025,335619,837999,2107947,5484373,14071891,36275323,95881995,

%U 250956301,659257445,1763642977,4685724391,12496708267,33766814039,90846586161,245197523769

%N a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^3) + 1 + 1/x^(k^3)).

%C All terms are odd.

%p b:= proc(n, i) option remember; `if`(n>(i*(i+1)/2)^2, 0,

%p `if`(i=0, 1, b(n, i-1)+b(n+i^3, i-1)+b(abs(n-i^3), i-1)))

%p end:

%p a:= n-> b(0, n):

%p seq(a(n), n=0..35); # _Alois P. Heinz_, Jan 21 2024

%t Table[Coefficient[Product[x^(k^3) + 1 + 1/x^(k^3), {k, 1, n}], x, 0], {n, 0, 33}]

%Y Cf. A000578, A007576, A158118, A350249.

%K nonn

%O 0,7

%A _Ilya Gutkovskiy_, Jan 20 2024