I'm stuck at finding the general term of the sequence $$1, 1, 1, 2, 6, 14, 28, 56, 118, 254, 541, 1140, 2401, 5074, \ldots$$ According to OEIS, Colin Barker conjectured the recurrence relation to be $$2\cdot a_{n-1}-a_{n-2}+2\cdot a_{n-3}+a_{n-4}+a_{n-5}-a_{n-7}-a_{n-8}$$ and the generating function $$-\frac{x\cdot (x^3+x-1)}{x^8+x^7-x^5-x^4-2\cdot x^3+x^2-2\cdot x+1}$$ I found the power series of the generating function to be $$x + x^2 + x^3 + 2 x^4 + 6 x^5 + O(x^5)$$ but I don't know how to proceed. Moreover I would like to find a closed form formula to get the general $a_n^3$ term, but I believe once I get the general formula for $a_n$ I am done, since $a_n$ has a linear recurrence recurrence implies that also $a_n^3$ has one.
Thanks in advance!