I was inspired by Euler's pentagonal number theorem to play with some products so I began evaluating
$$ \prod_{k=1}^{\infty} \left[ 1+\frac{-1+i\sqrt{3}}{2}x^k + \frac{-1-i\sqrt{3}}{2}x^{2k}\right] = \sum_{k=0}^{\infty} b_k x^k$$
It's relatively easy to see why the product on the left hand side results in some massive cancellation so I evaluated it to some pretty high terms and here is so far what the first 11 terms of the infinite sum look like, ie sum $\sum_{k=0}^{11} b_k x^k$ equals
$$ 1+ \left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)x-x^2-x^3+ \left(-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)x^4 + \left(\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)x^5+ \left(-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)x^6 + \left(\frac{3}{2}-\frac{i \sqrt{3}}{2}\right)x^7+x^8+x^9+x^{10}+ \left(\frac{3}{2}+\frac{i \sqrt{3}}{2}\right)x^{11} ... $$
Clearly these terms are relatively bounded (which we expect from the cancellation), but it's not really clear what the pattern is to these coefficients. They dont even stay constrained to 6th roots of unity as the $x^7$ term and the $x^{11}$ start to drift off the unit circle. I guess I wanted to ask for some help if anyone can notice any kind of pattern in these coefficients/subset of the coefficients. All 50 terms I have computed so far are presented below:
(Apologies for the weird ordering of coefficients, I just can't get mathematica to do what i need it to do) $$ \left(-19+5 i \sqrt{3}\right) x^{50}+\left(-15+7 i \sqrt{3}\right) x^{49}+\left(-\frac{25}{2}+\frac{13 i \sqrt{3}}{2}\right) x^{48}+\left(-10+8 i \sqrt{3}\right) x^{47}+\left(-6+8 i \sqrt{3}\right) x^{46}+\left(-5+8 i \sqrt{3}\right) x^{45}+\left(-\frac{3}{2}+\frac{15 i \sqrt{3}}{2}\right) x^{44}+\left(\frac{1}{2}+\frac{15 i \sqrt{3}}{2}\right) x^{43}+\left(4+7 i \sqrt{3}\right) x^{42}+\left(\frac{11}{2}+\frac{13 i \sqrt{3}}{2}\right) x^{41}+\left(7+5 i \sqrt{3}\right) x^{40}+\left(9+4 i \sqrt{3}\right) x^{39}+\left(10+2 i \sqrt{3}\right) x^{38}+\left(9+i \sqrt{3}\right) x^{37}+\left(\frac{17}{2}-\frac{i \sqrt{3}}{2}\right) x^{36}+\left(7-i \sqrt{3}\right) x^{35}+\left(7-2 i \sqrt{3}\right) x^{34}+\left(\frac{11}{2}-\frac{5 i \sqrt{3}}{2}\right) x^{33}+\left(\frac{7}{2}-\frac{7 i \sqrt{3}}{2}\right) x^{32}+\left(1-3 i \sqrt{3}\right) x^{31}+\left(1-3 i \sqrt{3}\right) x^{30}+\left(-\frac{3}{2}-\frac{1}{2} 5 i \sqrt{3}\right) x^{29}+\left(-\frac{3}{2}-\frac{1}{2} 3 i \sqrt{3}\right) x^{28}+\left(-2-i \sqrt{3}\right) x^{27}+\left(-1-i \sqrt{3}\right) x^{26}+\left(-\frac{5}{2}-\frac{i \sqrt{3}}{2}\right) x^{25}-x^{24}+\left(-\frac{3}{2}-\frac{i \sqrt{3}}{2}\right) x^{23}+\left(-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right) x^{22}+\left(-1-i \sqrt{3}\right) x^{21}-2 x^{20}-2 x^{19}+\left(-\frac{3}{2}+\frac{i \sqrt{3}}{2}\right) x^{18}+\left(-\frac{3}{2}+\frac{i \sqrt{3}}{2}\right) x^{17}+\left(-1+i \sqrt{3}\right) x^{16}+\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right) x^{15}+i \sqrt{3} x^{14}+\left(\frac{1}{2}+\frac{i \sqrt{3}}{2}\right) x^{13}+i \sqrt{3} x^{12}+\left(\frac{3}{2}+\frac{i \sqrt{3}}{2}\right) x^{11}+x^{10}+x^9+x^8+\left(\frac{3}{2}-\frac{i \sqrt{3}}{2}\right) x^7+\left(-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right) x^6+\left(\frac{1}{2}-\frac{i \sqrt{3}}{2}\right) x^5+\left(-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right) x^4-x^3-x^2+\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right) x+1 $$