These can be solved with basic algebra, but the process is long and tedious.
Don't know anyone who'd want to do it . . . Okay, it's me.
Take the differences of consecutive terms in the Partial Sums sequence.
Then take the differences of the differences . . . and so on.
We see that the 4th differences are constant.
This tells us that the generating function is of the 4th degree.
The general quartic function is: .
Use the first 5 terms of the sequence and set up this system:
Solve the system: .
. . Hence: .
This can be drastically simplified . . .
. . . .