 Negative-binomial sum of exponential claims

Let $p+q=1$ and let the original independent count have negative binomial distribution with probability mass function $\left(\genfrac{}{}{0ex}{}{n+k-1}{k}\right)p^n{q}^k$ on $k\ge 0$. With unit-rate exponential distribution severities, its moment-generating function is $[p+qp/(p-t){]}^n$ for $t<p$. Thus it also has a random sum of independent claims representation with independent binomial distribution count $B$ and rate-$p$ exponential distribution steps. For $n=2$ the law is $p^2{\delta }_0+2pq\mathrm{Exp}(p)+q^2\mathrm{Erlang}(2,p)$. The hurdle decomposition of a positive random sum therefore has zero mass $p^2$ and positive density $p^2{e}^{-px}(2q+q^{2}x)/(1-p^2)$.