Convolve {1, 2, 1} and {1, 2, 3, 1}

Given $h[n]=\{1,2,3,1\}$ with $h[0]$ = first 1, so $n=0..3$. No origin is marked for $x[n]$, so take $x[0]$ = first sample: $x[n]=\{1,2,1\}$, $n=0..2$.

Output: $y[n]=\sum_k x[k]h[n-k]$, from $n=0$ to $5$ (length $3+4-1=6$). Since $x[n]=\delta[n]+2\delta[n-1]+\delta[n-2]$, $y[n]=h[n]+2h[n-1]+h[n-2]$.

Tabular method

$$\begin{aligned} y[2]&=x[0]h[2]+x[1]h[1]+x[2]h[0]=3+4+1=8\\ y[3]&=x[0]h[3]+x[1]h[2]+x[2]h[1]=1+6+2=9 \end{aligned}$$

Sum check: $\sum y=28=(\sum x)(\sum h)=4\times7$.

Answer: $y[n]=\{1,\ 4,\ 8,\ 9,\ 5,\ 1\}$ for $n=0,\dots,5$ (↑ at the first 1). If $x[n]$ were taken with origin at its middle sample (2), the same values would simply start at $n=-1$.