Convolution of delta function in discrete time with parameter
You can use the following argumentation to find the result. The discrete time unit-sample function $\delta[n]$ has the following property for integer $M$:
$$ \delta[Mn] = \delta[n] $$ and more generally you can conlcude that for integer $M$ and $d$ we have
$$ \delta[M(n-d)] = \delta[n-d] $$

Therefore you can replace $\delta[5n-20] = \delta[5(n-4)]$ with $\delta[n-4]$ and proceed as usual to find the result of the convolution as:

$$y[n] = n\delta(5n-20) \star 3^{5n}u[n] = n\delta(n-4) \star 3^{5n}u[n] = 4 \cdot 3^{5(n-4)}u[n-4] $$

where in this last line we have used the sifting and shifting properties of the impulse function.