confused in difference between calculations of convolution and multiplication?
Just write down the convolution sum to see what's going on:

$$y[n]=\sum_{k=-\infty}^{\infty}x[k]p[n-k]\tag{1}$$

where we define the elements of the sequences $x[n]$ and $p[n]$ as equal to zero for the index $n$ outside the range of non-zero values, i.e., outside $n\in[0,2]$. Taking into account those zero values, we can rewrite $(1)$ with finite summation limits:

$$y[n]=\sum_{k=0}^{n}x[k]p[n-k]\tag{2}$$

because for $k&lt;0$ all $x[k]$ are zero, and for $k&gt;n$ the elements $p[n-k]$ are zero. Now you see that as $n$ increases from $0$ to  $2$ (outside that range the term $x[k]$ will be zero), you'll get more terms in the sum $(2)$:

$$\begin{align}y[0]&amp;=x[0]\cdot p[0]\\
y[1]&amp;=x[0]\cdot p[1]+x[1]\cdot p[0]\\
y[2]&amp;=x[0]\cdot p[2]+x[1]\cdot p[1]+x[2]\cdot p[0]\end{align}$$