The distribution of the n$n$-th convolution of the Laplace distribution can be computed from the characteristic function: exp(i mu t)/(1+b^2 t^2). (See the Wikipedia articlesee on Wikipedia). The: $$\frac{\exp(i \mu t)}{1+b^2 t^2} \,.$$ The characteristic function of the n$n$-th convolution becomes: exp(i n mu t)/(1+b^2 t^2)n = exp(i n mu t)/((1 - i b t)^n ((1 + i b t)^n) .$$\frac{\exp(i n \mu t)}{(1+b^2 t^2)^n} = \frac{\exp(i n \mu t)}{(1 - i b t)^n (1 + i b t)^n} \,.$$ The inverse Fourier transform can be computed using the residue theorem. The integration contour is closed from the upper or lower half plane according to the sign of (x-n mu)$(x-n \mu)$.
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