Convolution of Input signal and Impulse response I can solve the output if the input is 

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

and the impulse response is 

$$ h[n]= 3\delta[n] − 2 \delta[n−1] − \delta[n−2] + \delta[n−3]$$

I calculate the convolution between the strings [1 1 2 1]  and [3 -2 -1 1].

But I got a problem in computing the output when the input is 
$$ x[n] = (\frac{1}{4})^{-n}  \left( \delta[n +1]  − \delta[n −4] \right)$$

and the impulse response is

$$h[n] = \delta[n] −\delta[n −5].$$

I took different strings but actually could not find what should be input and impulse response string and why?
In the second case, which caused you the trouble, you should first use the sifting property of the impulse to find out the input signal $x[n]$ and then proceed with convolution.

sifting property is: 
$$ f[n] \delta[n-d] = f[d]\delta[n-d]$$

Apply this to simplify your input signal $x[n]$ as follows:

\begin{align}
x[n] &amp;= (\frac{1}{4})^{-n} \left(\delta[n+1] - \delta[n-4] \right) \\
&amp; = (\frac{1}{4})^{-n} \delta[n+1] - (\frac{1}{4})^{-n} \delta[n-4] \\
&amp; = (\frac{1}{4})^{-(-1)} \delta[n+1] - (\frac{1}{4})^{-(4)} \delta[n-4] \\
x[n] &amp; =  \frac{1}{4} \delta[n+1] - 256 ~ \delta[n-4] \\
\end{align}

So, your input sequence is $x[n] = \frac{1}{4} \delta[n+1] - 256 ~\delta[n-4] $

Convolving this with the impulse response of 
$$h[n] = \delta[n] - \delta[n-5]$$ is easy, this time by utilizing the shifting property of the impulse which is:

$$ f[n] \star \delta[n-d] = f[n-d] $$

Where $\star$ is th convolution. Then apply distribution of convolution over addition to get your result as:

\begin{align}
y[n] &amp;= x[n] \star h[n] \\
&amp; = x[n] \star (\delta[n] - \delta[n-5] ) \\
&amp;= x[n] - x[n-5] \\
&amp;= (\frac{1}{4} \delta[n+1] - 256 ~\delta[n-4]) - (\frac{1}{4} \delta[n-4] - 256~ \delta[n-9]) \\
&amp;= \frac{1}{4} \delta[n+1] - 256~ \delta[n-4] - \frac{1}{4} \delta[n-4] + 256~ \delta[n-9] \\
y[n] &amp;= \frac{1}{4} \delta[n+1] - (256~ \delta[n-4] + \frac{1}{4}) \delta[n-4] + 256~ \delta[n-9] \\
\end{align}

I hope this helps.