Convolve {1, 1, 2, −1} and {1, 2}

Given $h[n]=\{1,1,2,-1\}$ with $h[0]=1$ (first), so $n=0..3$, and $x[n]=\{1,2\}$ with $x[0]=1$, so $n=0,1$.

Since $x[n]=\delta[n]+2\delta[n-1]$,

$$y[n]=x[n]*h[n]=h[n]+2h[n-1]$$

Output length $=2+4-1=5$, for $n=0$ to $4$.

Tabular method

$$\begin{aligned} y[0]&=1,\quad y[1]=1+2(1)=3,\quad y[2]=2+2(1)=4\\ y[3]&=-1+2(2)=3,\quad y[4]=2(-1)=-2 \end{aligned}$$

Sum check: $\sum y=9=(\sum x)(\sum h)=3\times3$.

Answer: $y[n]=\{1,\ 3,\ 4,\ 3,\ -2\}$ for $n=0,\dots,4$, with $y[0]=1$.

Sketch of y[n]

    y[n]
      4         o
      3     o   |   o
      1 o   |   |   |
      0 +---+---+---+---+--> n
     -2                 o
        0   1   2   3   4