> 
> Find the formula for the convolution of the sequences:
> 
> 
> $a\_n=\begin {cases} 1 & 0\leq n \leq 4 \\ 0 & n \geq 5 \end{cases} $
> 
> 
> $b\_n = 1$ $\forall n \in \mathbb{N}$
> 
> 
> 

What I've been doing:

In other words:

$a\_n = 1,1,1,1,1,0,0,...$ and $b\_n=1,1,1,1,...$

So that means that their generating functions are:

$a(x)=1+x+x^2+x^3+x^4=\sum \_{i=0} ^{4} x^i$

$b(x)=1+x+x^2+x^3+...=\sum \_{j=0} ^{\infty} x^i$

So using the convolution formula i.e: $b(x)a(x)=\sum \_{j=0} ^{\infty}(\sum \_{i=0} ^{4}a\_ib\_{j-i})x^4=\sum \_{j=0} ^{\infty}(\sum \_{i=0} ^{4}x^i(x^{i-j}))x^4=\sum \_{j=0} ^{\infty}(\sum \_{i=0} ^{4}x^{2i-j})x^4$

And I can't get past that nor do I know if what I did was correct... Can someone guide me?