Not sure I understand your problem exactly, but I'll try to give you a closed form equation for the recursive relation

So we can get

Taking sums of all sides we get

Let

The left side of our equation is

The first term on the right is

The second term on the right is

So since you said that we have

According to wolframalpha.com the right side becomes

So

And by power series representations we have that

Thus the coefficient of is

And now that I've done all this, I realize that induction clearly shows that every *note after re-reading the problem you may not have given the first 4 values of the sequence, which makes this wrong

Eh well... at least I got to review working with generating functions. You can change one of your initial values or many of them and repeat this analysis so maybe it will be helpful