This is a question that I posted in physics stackexchange, but I believe the problem is purely mathematical. Original question here.
One can write pressure of an interacting dilute gas as an expansion in its fugacity $\xi = e^{\beta \mu}$ as,
$$\frac{P}{n_0T} = \sum_{l =1}^\infty b_{l} \xi^{l}$$
From this we can calculate the density as, $$\frac{n}{n_0} = \sum_{l=1}^\infty l b_l \xi^l$$
On the other hand we have the virial expansion, $$\frac{P}{nT} = \sum_{l=0}^\infty a_l \left(\frac{n}{n_0}\right)^{l-1}$$
$a_l$ and $b_l$ are related to each other which can be found from matching the above two relation in each order of fugacity. If one take $b_1 = 1$ then we have $a_1 = 1$, $a_2 = -b_2$, $a_3 = 2(2b_2^2 - b_3)$ and so on. Is there a simpler way to calculate this? For example is there a recursive formula to find the next coefficients from the previous one?