<s>[INST] Find the coefficients of product Given the following product,
$$(1+ax)(1+a^2x)(1+a^3x)\cdots (1+a^mx) $$
where $a$ is some real number which will be taken to be unity in the end. I want to know the coefficient of general term of expansion $x^p$. I know that the coefficient of $x^p$ comes from choosing $p$ terms from $m$, but the terms will reduce because some kinds of products have the same coefficients. How to write the coefficient step by step?  Thanks in advance.
 [/INST] Products of this type are known to be generalisations of the binomial
\begin{align*}
(1+x)^m=\sum_{k=0}^m\binom{m}{k}x^k
\end{align*}
Often such generalisations are written with $q$ instead of $a$ and here I'll follow this convention. We denote with $[x^p]$ the coefficient of $x^p$ in a series and want to find
\begin{align*}
[x^p]\left(1+qx\right)\left(1+q^2x\right)\cdots\left(1+q^mx\right)=[x^p]\prod_{k=1}^m\left(1+q^kx\right)\tag{1}
\end{align*}
According to the q-binomial theorem the following is valid
\begin{align*}
\color{blue}{\prod_{k=0}^{m-1}\left(1+q^kx\right)=\sum_{k=0}^mq^{\binom{k}{2}}\binom{m}{k}_q x^k}\tag{2}
\end{align*}
with $\binom{m}{k}_{q}$ the so-called q-binomial coefficients. The q-binomial coefficients follow a recurrence relation similar to the binomial coefficients, namely
\begin{align*}
\binom{m}{k}_q&=q^k\binom{m-1}{k}+\binom{m-1}{k-1}_q\qquad\qquad 1\leq k\leq m\tag{3.1}\\
\binom{m}{0}_q&=\binom{m}{m}_q=1\tag{3.2}\\
\binom{m}{1}_q&=\binom{m}{m-1}_q=1+q+\cdots+q^{k-1}\tag{3.3}
\end{align*}
We can now transform (1) to use the sum formula in (2) so that we can extract the coefficient of $x^p$.

We obtain from (1) and (2)
\begin{align*}
\color{blue}{[x^p]}\color{blue}{\prod_{k=1}^m\left(1+q^kx\right)}
&=[x^p]\prod_{k=1}^m\left(1+q^{k-1}(qx)\right)\tag{4.1}\\
&=[x^p]\prod_{k=0}^{m-1}\left(1+q^k(qx)\right)\tag{4.2}\\
&=[x^p]\sum_{k=0}^mq^{\binom{k}{2}}\binom{m}{k}_q (qx)^k\tag{4.3}\\
&=[x^p]\sum_{k=0}^mq^{\binom{k+1}{2}}\binom{m}{k}_q x^k\tag{4.4}\\
&\,\,\color{blue}{=q^{\binom{p+1}{2}}\binom{m}{p}_q}\tag{4.5}
\end{align*}

Comment:

*

*In (4.1) we write $q^k=q^{k-1}q$.


*In (4.2) we shift the index to start with $k=0$ and are now able to apply (2).


*In (4.3) we apply (2), replacing $x$ with $qx$.


*In (4.4) we use $\binom{q}{2}+q=\frac{1}{2}q(q-1)+q=\frac{1}{2}q(q+1)=\binom{q+1}{2}$.


*In (4.5) we select the coefficient of $x^p$.
Example $m=5, p=3$: Let's calculate a small example. We obtain with $m=5$ and $p=3$:
\begin{align*}
\color{blue}{[}&\color{blue}{x^3]\prod_{k=1}^5\left(1+q^kx\right)}=q^{\binom{4}{2}}\binom{5}{2}_q\tag{$\rightarrow 4.1, 4.5$}\\
&=q^6\left(q^2\binom{4}{2}_q+\binom{4}{1}_q\right)\tag{$\rightarrow 3.1$}\\
&=q^6\left(q^2\left(q^2\binom{3}{2}_q+\binom{3}{1}_q\right)+\binom{4}{1}_q\right)\tag{$\rightarrow 3.1$}\\
&=q^6\left(q^4\binom{3}{1}_q+q^2\binom{3}{1}_q+\binom{4}{1}_q\right)\tag{$\rightarrow 3.3$}\\
&=q^6\left(q^4\left(1+q+q^2\right)+q^2\left(1+q+q^2\right)+\left(1+q+q^2+q^3\right)\right)\\
&\,\,\color{blue}{=q^6\left(1+q+2q^2+2q^3+2q^4+q^5+q^6\right)}
\end{align*}
in accordance with a plausibility check by WolframAlpha.
</s>