I am having this sequence: $a(n) = 2\cdot a(n-1) - a(n-2) + 2 \cdot a(n-3) + a(n-4) + a(n-5) - a(n-7) - a(n-8)$ with this generating function: $x \cdot (1 - x - x^3) / (1 - 2 \cdot x + x^2 - 2 \cdot x^3 - x^4 - x^5 + x^7 + x^8)$
Is it possible to find an expression for the sum $a(1)^3 + a(2)^3 + a(3)^3 + \cdots + a(n)^3$?
I know the expression of $1^3+2^3+3^3+ \cdots \;$, but how can I "put" the other sequence in this sequence? Should I multiply the generating functions and try to do some "tricks" for evaluating the sum? Or maybe plug in the expression of the generating function to the expression of the other generating function?
Too long for a comment (which, in any manner, does not try to answer the question).
This is sequence $A249665$ in $OEIS$ (have a look here and it seems to be quite recent (year $2018$).
The list of the first $a_n$ is $$\{1,1,1,2,6,14,28,56,118,254,541,1140,2401,5074,10738,22711,48001,101447,214446\}$$
Now, if you want to compute $$S_p=\sum_{n=1}^p a_n^3$$ I assume (and hope !) that you enjoy big numbers since $$\left( \begin{array}{cc} p & S_p \\ 1 & 1 \\ 2 & 2 \\ 3 & 3 \\ 4 & 11 \\ 5 & 227 \\ 6 & 2971 \\ 7 & 24923 \\ 8 & 200539 \\ 9 & 1843571 \\ 10 & 18230635 \\ 11 & 176571056 \\ 12 & 1658115056 \\ 13 & 15499402257 \\ 14 & 146131947481 \\ 15 & 1384273214753 \\ 16 & 13098369026184 \\ 17 & 123697281170185 \\ 18 & 1167738453611808 \\ 19 & 11029485294400344 \\ 20 & 104207881192114219 \\ 21 & 984513792009594094 \\ 22 & 9300131757294325441 \\ 23 & 87850531055100094566 \\ 24 & 829875844095743052222 \\ 25 & 7839544341677224723097 \\ 26 & 74057216493851207940236 \\ 27 & 699585106377423299348164 \\ 28 & 6608651388350619132919300 \\ 29 & 62428935435804168232048591 \\ 30 & 589738667341010510077068047 \end{array} \right)$$ which seem to be quite linear $(R^2=0.999991)$ in a logarithmic scale $$\log(S_p)=a + b p$$
$$\begin{array}{clclclclc} \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\ a & -5.53320 & 0.08120 & \{-5.69435,-5.37205\} \\ b & +2.24276 & 0.00140 & \{+2.23999,+2.24553\} \\ \end{array}$$