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Within the approximation that a factory has either zero or one accident, let $N$ be the accident count. The factories' independence gives $N\sim \mathrm{Binomial}(m,p)$. Given an accident, its cluster of claims has a Poisson distribution of mean $\lambda$, independently across factories. Its probability generating function is $G_X(z)=e^{\lambda (z-1)}$. Thus this is a binomial accident portfolio with Poisson clusters, withA factory can contribute no claims either because it has no accident or because its accident's Poisson cluster is empty. Hence the no-claim probability differs from the no-accident probability $q^m$.

For $0\le n\le m$, the accident probability mass function is $p_n=\left(\genfrac{}{}{0ex}{}{m}{n}\right)p^n{q}^{m-n}$. For $1\le n\le m$,Therefore its Panjer claim-count class parameters areAt $n=m+1$ the multiplier is zero, and afterwards both successive masses are zero, so the recurrence holds for all $n\ge 1$. Finally, $f_0=e^{-\lambda }$ and $G_N(z)=(q+pz{)}^m$ reproduce the same initialization $g_0=G_N(f_0)$.