# Finite series within finite series

## Homework Statement

I need to find a closed form for what at first light would be a straightforward finite series. Calculating it explicitly to a particular degree is not difficult, but I just can't find the closed form for the general case.

## Homework Equations

For $$N>m$$, the series is:
$$\sum_{k_m=1}^{N}(b^{m-1}x)^{k_m}\sum_{k_{m-1}=1}^{k_m}(b^{m-2}x)^{k_{m-2}}\cdots \sum_{k_2=1}^{k_3}(b x)^{k_2}\sum_{k_1=1}^{k_2}x^{k_2}$$

## The Attempt at a Solution

I know the answer will look something like this:
$$\sum_{i=0}^{m}a_i(b,x) x^{i N}$$
where the $$a_i(b,x)$$ are linear combinations of b's and $$\frac{1}{1-(b^p x^r)^{-1}}$$, but I just can't find the general expression. Even a recurrence relation would be sufficient for my needs.

Any help will be greatly appreciated.