The Generating Function for your coefficients is
g[x_] := (x - x^2 - x^4)/(1 - 2 x + x^2 - 2 x^3 - x^4 - x^5 + x^7 + x^8)
Your coefficients then are obtained by (which is sort of a symbolic expression)
a[n_] := 1/n! Derivative[n][g][0]
e.g.
In[25]:=aa= a /@ Range[30]
Out[25]= {1, 1, 1, 2, 6, 14, 28, 56, 118, 254, 541, 1140, 2401, 5074, \
10738, 22711, 48001, 101447, 214446, 453355, 958395, 2025963, \
4282685, 9053286, 19138115, 40456779, 85522862, 180789396, 382176531, \
807895636}
But this as well gives only numerical values, which might be calculated quicker by recursion.
As check
With[{n = 15},
{aa[[n]],2 aa[[n - 1]] - aa[[n - 2]] + 2 aa[[n - 3]] + aa[[n - 4]] + aa[[n - 5]] - aa[[n - 7]] - aa[[n - 8]]}]
Unfortunately that doesn't give a reasonable acces to a[n]^3 and its sum.